[PDF] TRIGONOMETRIC IDENTITIES AND EQUATIONS

1 Chapter 7 Unit Trigonometry (Chapters 5 8) TRIGONOMETRIC IDENTITIES ND EQUTIONS CHPTER OBJECTIVES Use reciprocal, quot...

Chapter

Unit 2 Trigonometry (Chapters 5–8)

TRIGONOMETRIC IDENTITIES AND EQUATIONS

CHAPTER OBJECTIVES • • • • • •

420

Chapter 7

Use reciprocal, quotient, Pythagorean, symmetry, and opposite-angle identities. (Lesson 7-1) Verify trigonometric identities. (Lessons 7-2, 7-3, 7-4) Use sum, difference, double-angle, and half-angle identities. (Lessons 7-3, 7-4) Solve trigonometric equations and inequalities. (Lesson 7-5) Write a linear equation in normal form. (Lesson 7-6) Find the distance from a point to a line. (Lesson 7-7)

Trigonometric Identities and Equations

Many sunglasses have polarized lenses that reduce the intensity of light. When unpolarized light passes through a polarized p li c a ti lens, the intensity of the light is cut in half. If the light then passes through another polarized lens with its axis at an angle of to the first, the intensity of the light is again diminished. OPTICS

• Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities, and opposite-angle identities.

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OBJECTIVE

Basic Trigonometric Identities R

7-1

Axis 1 Lens 2 Unpolarized light

Lens 1 Axis 2

The intensity of the emerging light can be found by using the formula I csc

I I0 0 2 , where I0 is the intensity of the light incoming to the second polarized lens, I is the intensity of the emerging light, and is the angle between the axes of polarization. Simplify this expression and determine the intensity of light emerging from a polarized lens with its axis at a 30° angle to the original. This problem will be solved in Example 5.

In algebra, variables and constants usually represent real numbers. The values of trigonometric functions are also real numbers. Therefore, the language and operations of algebra also apply to trigonometry. Algebraic expressions involve the operations of addition, subtraction, multiplication, division, and exponentiation. These operations are used to form trigonometric expressions. Each expression below is a trigonometric expression. cos x x

sin2 a cos2 a

1 sec A tan A

A statement of equality between two expressions that is true for all values of the variable(s) for which the expressions are defined is called an identity. For example, x2 y2 (x y)(x y) is an algebraic identity. An identity involving trigonometric expressions is called a trigonometric identity. If you can show that a specific value of the variable in an equation makes the equation false, then you have produced a counterexample. It only takes one counterexample to prove that an equation is not an identity. Lesson 7-1

Basic Trigonometric Identities

421

Example

1 Prove that sin x cos x tan x is not a trigonometric identity by producing a counterexample. 4

Suppose x . sin x cos x tan x 4 4 2 2 1 2 2 1 1 2

sin cos tan Replace x with .

Since evaluating each side of the equation for the same value of x produces an inequality, the equation is not an identity.

Although producing a counterexample can show that an equation is not an identity, proving that an equation is an identity generally takes more work. Proving that an equation is an identity requires showing that the equality holds for all values of the variable where each expression is defined. Several fundamental trigonometric identities can be verified using geometry. y

Recall from Lesson 5-3 that the trigonometric functions can be defined using the unit circle. From y 1 the unit circle, sin , or y and csc . That is, 1 y 1 sin . Identities derived in this manner are

(x, y) 1

csc

called reciprocal identities.

The following trigonometric identities hold for all values of where each expression is defined. Reciprocal Identities

1 csc 1 csc sin 1 tan cot

sin

1 sec 1 sec cos 1 cot tan

cos

sin

Returning to the unit circle, we can say that tan . This is an cos x example of a quotient identity.

Quotient Identities

422

Chapter 7

The following trigonometric identities hold for all values of where each expression is defined.

Trigonometric Identities and Equations

sin tan cos

cos cot sin

Since the triangle in the unit circle on the previous page is a right triangle, we may apply the Pythagorean Theorem: y2 x 2 12, or sin2 cos2 1. Other identities can be derived from this one. sin2 cos2 1 sin2 1 cos2 2 cos cos2 cos2

tan2 1 sec2

Divide each side by cos2 . Quotient and reciprocal identities

Likewise, the identity 1 cot2 csc2 can be derived by dividing each side of the equation sin2 cos2 1 by sin2 . These are the Pythagorean identities.

Pythagorean Identities

The following trigonometric identities hold for all values of where each expression is defined. sin2 cos2 1

tan2 1 sec2

1 cot2 csc2

You can use the identities to help find the values of trigonometric functions.

Example

2 Use the given information to find the trigonometric value. 3 2

a. If sec , find cos . 1 sec

cos

Choose an identity that involves cos and sec .

2 3 3 or Substitute for sec and evaluate. 3

4 3

b. If csc , find tan . Since there are no identities relating csc and tan , we must use two identities, one relating csc and cot and another relating cot and tan . csc2 1 cot2 Pythagorean identity

4 3

1 cot2 Substitute for csc .

16 1 cot2 9 7 cot2 9

7 cot 3

Take the square root of each side.

Now find tan . 1 cot

tan

Reciprocal identity

37 , or about 1.134 7

Lesson 7-1

Basic Trigonometric Identities

423

To determine the sign of a function value, you need to know the quadrant in which the angle terminates. The signs of function values in different quadrants are related according to the symmetries of the unit circle. Since we can determine the values of tan A, cot A, sec A, and csc A in terms of sin A and/or cos A with the reciprocal and quotient identities, we only need to investigate sin A and cos A.

Relationship between angles A and B

Ca s e 1

The angles differ by a multiple of 360°. B A 360k° or B A 360 k°

The angles differ by an odd multiple of 180°.

Diagram

The sum of the angles is a multiple of 360°.

Since A and A 360k° are coterminal, they share the same value of sine and cosine.

(a, b)

A 360k ˚

A 180˚ (2k 1) y

Since A and A 180°(2k 1) have terminal sides in diagonally opposite quadrants, the values of both sine and cosine change sign.

(a, b)

B A 180°(2k 1) or B A 180°(2k 1)

Conclusion

O (a, b)

Since A and 360k° A lie in vertically adjacent quadrants, the sine values are opposite but the cosine values are the same.

(a, b)

360k ˚ A

A B 360k° or B 360k° A

(a, b)

The sum of the angles is an odd multiple of 180°. A B 180° (2k 1) or B 180°(2k 1) A

(a, b) y 180˚ (2k 1) A

(a, b)

Since A and 180°(2k 1) A lie in horizontally adjacent quadrants, the sine values are the same but the cosine values are opposite.

These general rules for sine and cosine are called symmetry identities.

Symmetry Identities Case Case Case Case

1: 2: 3: 4:

The following trigonometric identities hold for any integer k and all values of A. sin sin sin sin

(A 360k°) sin A (A 180°(2k 1)) sin A (360k° A) sin A (180°(2k 1) A) sin A

cos cos cos cos

(A 360k°) cos A (A 180°(2k 1)) cos A (360k° A) cos A (180°(2k 1) A) cos A

To use the symmetry identities with radian measure, replace 180° with and 360° with 2. 424

Chapter 7

Trigonometric Identities and Equations

Example

3 Express each value as a trigonometric function of an angle in Quadrant I. a. sin 600° Relate 600° to an angle in Quadrant I. 600° 60° 3(180°)

600° and 60° differ by an odd multiple of 180°.

sin 600° sin (60° 3(180°)) Case 2, with A 60° and k 2 sin 60°

19 4

b. sin 19 4 19 5 4 4

20 4

The sum of and , which is or 5, is an odd multiple of .

19 4

sin sin 5 4

Case 4, with A and k 3

sin

c. cos (410°) The sum of 410° and 50° is a multiple of 360°. 410° 360° 50° cos (410°) cos (360° 50°)

Case 3, with A 50° and k 1

cos 50°

37 6

d. tan 37 and differ by a multiple of 2. 6 6 37 3(2) Case 1, with A and k 3 6 6 6 37 Rewrite using a quotient identity since the sin 6 symmetry identities are in terms of sine and cosine. 37 tan 37 6 cos 6

sin 3(2) 6 cos 3(2) 6 sin 6 or tan 6 cos 6

Quotient identity

Lesson 7-1

Basic Trigonometric Identities

425

Case 3 of the Symmetry Identities can be written as the opposite-angle identities when k 0.

OppositeAngle Identities

The following trigonometric identities hold for all values of A. sin (A) sin A cos (A) cos A

The basic trigonometric identities can be used to simplify trigonometric expressions. Simplifying a trigonometric expression means that the expression is written using the fewest trigonometric functions possible and as algebraically simplified as possible. This may mean writing the expression as a numerical value.

Examples

4 Simplify sin x sin x cot2 x. sin x sin x cot2 x sin x (1 cot2 x) Factor. sin x csc2 x

Pythagorean identity: 1 cot 2 x csc 2 x

1 sin x sin2 x 1 sin x

csc x

5 OPTICS Refer to the application at the beginning of the lesson. I csc

a. Simplify the formula I I0 0 2 .

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Reciprocal identity

Example

Reciprocal identity

b. Use the simplified formula to determine the intensity of light that passes through a second polarizing lens with axis at 30° to the original.

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I csc

a. I I0 0 2 I I0 I0 sin2

Reciprocal identity

I I0(1 sin2 )

Factor.

I I0 cos2

1 sin2 cos2

b. I I0 cos2 30°

3 I I0

3 4

I I0 The light has three-fourths the intensity it had before passing through the second polarizing lens.

426

Chapter 7

Trigonometric Identities and Equations

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Find a counterexample to show that the equation 1 sin x cos x is not an

identity. 2. Explain why the Pythagorean and opposite-angle identities are so named. 3. Write two reciprocal identities, one quotient identity, and one Pythagorean

identity, each of which involves cot . 4. Prove that tan (A) tan A using the quotient and opposite-angle identities. 5. You Decide

Claude and Rosalinda are debating whether an equation from their homework assignment is an identity. Claude says that since he has tried ten specific values for the variable and all of them worked, it must be an identity. Rosalinda explained that specific values could only be used as counterexamples to prove that an equation is not an identity. Who is correct? Explain your answer.

Guided Practice

Prove that each equation is not a trigonometric identity by producing a counterexample. 6. sin cos tan

7. sec2 x csc2 x 1

Use the given information to determine the exact trigonometric value. 2 8. cos , 0° 90°; sec 3 1 3 10. sin , ; cos 5 2

5 , ; tan 9. cot 2 2 4 11. tan , 270° 360°; sec 7

Express each value as a trigonometric function of an angle in Quadrant I. 7 12. cos 3

13. csc (330°)

Simplify each expression. csc 14. cot

15. cos x csc x tan x

16. cos x cot x sin x

17. Physics

When there is a current in a wire in a magnetic field, a force acts on the wire. The strength of the magnetic field can be determined using the formula F csc B , where F is the force on the wire, I is the current in the wire, is the I length of the wire, and is the angle the wire makes with the magnetic field. Physics texts often write the formula as F IB sin . Show that the two formulas are equivalent.

E XERCISES Prove that each equation is not a trigonometric identity by producing a counterexample.

Practice

18. sin cos cot

sec 19. sin tan

cos x 20. sec2 x 1 csc x

21. sin x cos x 1

22. sin y tan y cos y

23. tan2 A cot2 A 1

www.amc.glencoe.com/self_check_quiz

Lesson 7-1 Basic Trigonometric Identities

427

24. Find a value of for which cos cos cos . 2 2

Use the given information to determine the exact trigonometric value.

2 25. sin , 0° 90°; csc 5

3 , 0 ; cot 26. tan 2 4

1 27. sin , 0 ; cos 4 2

2 28. cos , 90° 180°; sin 3

11 29. csc , ; cot 2 3

5 30. sec , 90° 180°; tan 4

1 31. sin , 180° 270°; tan 3

2 3 32. tan , ; cos 3 2

7 33. sec , 180° 270°; sin 5

1 3 34. cos , 2; tan 8 2

4 35. cot , 270° 360°; sin 3

3 36. cot 8, 2; csc 2

sec2 A tan2 A 3 , find 37. If A is a second quadrant angle, and cos A . 4 2 sin2 A 2 cos2 A

Express each value as a trigonometric function of an angle in Quadrant I.

19 40. tan 5

27 39. cos 8 10 41. csc 3

42. sec (1290°)

43. cot (660°)

38. sin 390°

Simplify each expression.

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Applications and Problem Solving

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428

sec x 44. tan x

cot 45. cos

sin ( ) 46. cos ( )

47. (sin x cos x)2 (sin x cos x)2

48. sin x cos x sec x cot x

49. cos x tan x sin x cot x

50. (1 cos )(csc cot )

51. 1 cot2 cos2 cos2 cot2

sin x sin x 52. 1 cos x 1 cos x

53. cos4 2 cos2 sin2 sin4

54. Optics

Refer to the equation derived in Example 5. What angle should the axes of two polarizing lenses make in order to block all light from passing through?

55. Critical Thinking

Use the unit circle definitions of sine and cosine to provide a geometric interpretation of the opposite-angle identities.

Chapter 7 Trigonometric Identities and Equations

56. Dermatology

It has been shown that skin cancer is related to sun exposure. The rate W at which a person’s skin absorbs energy from the sun depends on the energy S, in watts per square meter, provided by the sun, the surface area A exposed to the sun, the ability of the body to absorb energy, and the angle between the sun’s rays and a line perpendicular to the body. The ability of an object to absorb energy is related to a factor called the emissivity, e, of the object. The emissivity can be calculated using the formula W sec e .

AS a. Solve this equation for W. Write your

answer using only sin or cos . b. Find W if e 0.80, 40°, A 0.75 m2, and S 1000 W/m2.

A skier of mass m descends a -degree hill at a constant speed. When Newton’s Laws are applied to the situation, the following system of equations is produced.

57. Physics

FN mg cos 0 mg sin k FN 0 where g is the acceleration due to gravity, FN is the normal force exerted on the skier, and k is the coefficient of friction. Use the system to define k as a function of . 58. Geometry

Show that the area of a regular polygon of n sides, each of 1 4

18n0°

length a, is given by A na2 cot . 59. Critical Thinking

The circle at the right is a unit CD are circle with its center at the origin. AB and tangent to the circle. State the segments whose measures represent the ratios sin , cos , tan , sec , cot , and csc . Justify your answers.

Mixed Review

y B

x F D

2 . (Lesson 6-8) 60. Find Cos1 2

61. Graph y cos x . (Lesson 6-5) 6 62. Physics A pendulum 20 centimeters long swings 3°30 on each side of its

vertical position. Find the length of the arc formed by the tip of the pendulum as it swings. (Lesson 6-1) 63. Angle C of ABC is a right angle. Solve the triangle if A 20° and c 35.

(Lesson 5-4) 64. Find all the rational roots of the equation 2x3 x2 8x 4 0. (Lesson 4-4) 65. Solve 2x2 7x 4 0 by completing the square. (Lesson 4-2) 66. Determine whether f(x) 3x 3 2x 5 is continuous or discontinuous at

x 5. (Lesson 3-5)

Extra Practice See p. A38.

Lesson 7-1 Basic Trigonometric Identities

429

67. Solve the system of equations algebraically. (Lesson 2-2)

x y 2z 3 4x y z 0 x 5y 4z 11 68. Write the slope-intercept form of the equation of the line that passes through

points at (5, 2) and (4, 4). (Lesson 1-4) B

69. SAT/ACT Practice Triangle ABC is inscribed in circle O,

and CD is tangent to circle O at point C. If mBCD 40°, find mA. A 60° B 50° C 40° D 30° E 20°

O C A

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CAREER OVERVIEW Degree Preferred: bachelor’s degree in engineering or a physical science

Related Courses: mathematics, geography, computer science, mechanical drawing

Outlook: slower than average through 2006

For more information on careers in cartography, visit: www.amc.glencoe.com

430

Chapter 7 Trigonometric Identities and Equations

While working on a mathematics assignment, a group of students p li c a ti derived an expression for the length of a ladder that, when held horizontally, would turn from a 5-foot wide corridor into a 7-foot wide corridor. They determined that the maximum length of a ladder that would fit was given by PROBLEM SOLVING

• Use the basic trigonometric identities to verify other identities. • Find numerical values of trigonometric functions.

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OBJECTIVES

Verifying Trigonometric Identities R

7-2

7 ft

7 sin 5 cos sin cos

() , where is the angle that the ladder makes with the outer wall of the 5-foot wide corridor. When their teacher worked the problem, she concluded that () 7 sec 5 csc . Are the two expressions for () equivalent? This problem will be solved in Example 2.

5 ft

Verifying trigonometric identities algebraically involves transforming one side of the equation into the same form as the other side by using the basic trigonometric identities and the properties of algebra. Either side may be transformed into the other side, or both sides may be transformed separately into forms that are the same.

Suggestions for Verifying Trigonometric Identities

• Transform the more complicated side of the equation into the simpler side. • Substitute one or more basic trigonometric identities to simplify expressions. • Factor or multiply to simplify expressions. • Multiply expressions by an expression equal to 1. • Express all trigonometric functions in terms of sine and cosine. You cannot add or subtract quantities from each side of an unverified identity, nor can you perform any other operation on each side, as you often do with equations. An unverified identity is not an equation, so the properties of equality do not apply.

Example

1 Verify that sec2 x tan x cot x tan2 x is an identity. Since the left side is more complicated, transform it into the expression on the right. sec2 x tan x cot x tan2 x 1 tan x

sec2 x tan x tan2 x sec2 x 1 tan2 x x 1 1 tan2 x tan2 x tan2 x

tan2

1 tan x

cot x Multiply. sec2 x tan2 x 1 Simplify.

We have transformed the left side into the right side. The identity is verified.

Lesson 7-2

Verifying Trigonometric Identities

431

7 sin 5 cos sin cos

that 7 sec 5 csc is an identity.

Examples 2 PROBLEM SOLVING Verify that the two expressions for () in the application at the beginning of the lesson are equivalent. That is, verify al Wo p li c a ti

Begin by writing the right side in terms of sine and cosine. 7 sin 5 cos 7 sec 5 csc sin cos 7 sin 5 cos 7 5 sin cos cos sin

sec , csc

7 sin 5 cos 7 sin 5 cos sin cos sin cos sin cos

Find a common denominator.

7 sin 5 cos 7 sin 5 cos sin cos sin cos

Simplify.

1 cos

1 sin

The students and the teacher derived equivalent expressions for (), the length of the ladder.

sin A

cos A

csc2 A cot2 A is an identity. 3 Verify that csc A sec A

Since the two sides are equally complicated, we will transform each side independently into the same form. sin A cos A csc2 A cot2 A csc A sec A 2 2 cos A sin A (1 cot A) cot A 1 1 cos A sin A

sin2 A cos2 A 1

Quotient identities; Pythagorean identity Simplify.

sin2 A cos2 A 1

The techniques that you use to verify trigonometric identities can also be used to simplify trigonometric equations. Sometimes you can change an equation into an equivalent equation involving a single trigonometric function.

Example

cot x

2. 4 Find a numerical value of one trigonometric function of x if cos x

You can simplify the trigonometric expression on the left side by writing it in terms of sine and cosine. cot x 2 cos x cos x sin x 2 cos x cos x 1 2 sin x cos x 432

Chapter 7

cos x cot x sin x

Definition of division

Trigonometric Identities and Equations

1 2 sin x

Simplify. 1 sin x

csc x 2 csc x cot x cos x

Therefore, if 2, then csc x 2.

You can use a graphing calculator to investigate whether an equation may be an identity.

GRAPHING CALCULATOR EXPLORATION ➧ Graph both sides of the equation as two

separate functions. For example, to test sin2 x (1 cos x)(1 cos x), graph y1 sin2 x and y2 (1 cos x)(1 cos x) on the same screen. ➧ If the two graphs do not match, then the

TRY THESE Determine whether each equation could be an identity. Write yes or no. 1. sin x csc x sin2 x cos2 x 2. sec x csc x 1 1 csc x sec x

3. sin x cos x

equation is not an identity. ➧ If the two sides appear to match in every

window you try, then the equation may be an identity.

WHAT DO YOU THINK? 4. If the two sides appear to match in every window you try, does that prove that the equation is an identity? Justify your answer. sec x cos x

5. Graph the function f(x) . tan x What simpler function could you set equal to f(x) in order to obtain an identity?

[, ] scl:1 by [1, 1] scl:1

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Write a trigonometric equation that is not an identity. Explain how you know it is

not an identity. 2. Explain why you cannot square each side of the equation when verifying a

trigonometric identity. 3. Discuss why both sides of a trigonometric identity are often rewritten in terms of

sine and cosine. Lesson 7-2 Verifying Trigonometric Identities

433

4. Math

Journal Create your own trigonometric identity that contains at least three different trigonometric functions. Explain how you created it. Give it to one of your classmates to verify. Compare and contrast your classmate’s approach with your approach.

Guided Practice

Verify that each equation is an identity. cot x 5. cos x csc x

1 cos x 6. tan x sec x sin x 1

1 7. csc cot csc cot

8. sin tan sec cos

9. (sin A cos A)2 1 2 sin2 A cot A

Find a numerical value of one trigonometric function of x. 1 10. tan x sec x 4

11. cot x sin x cos x cot x

12. Optics

The amount of light that a source provides to a surface is called the illuminance. The illuminance E in foot candles on a surface that is R feet from a source of light with intensity

Perpendicular to surface

I cos R

I candelas is E 2 , where

is the measure of the angle between the direction of the light and a line perpendicular to the surface being illuminated. Verify that I cot R csc

E 2 is an equivalent formula.

E XERCISES Practice

Verify that each equation is an identity.

sec A 13. tan A csc A

14. cos sin cot

1 sin x 15. sec x tan x cos x

1 tan x 16. sec x sin x cos x 2 sin2 1 18. sin cos sin cos

17. sec x csc x tan x cot x 2 sec A csc A 19. (sin A cos A)2 sec A csc A cos y 1 sin y 21. 1 sin y cos y cot2 x 23. csc x 1 csc x 1 25. sin cos tan cos2 1

20. (sin 1)(tan sec ) cos 22. cos cos () sin sin () 1 24. cos B cot B csc B sin B 1 cos x 26. (csc x cot x)2 1 cos x

cos x sin x 27. sin x cos x 1 tan x 1 cot x 28. Show that sin cos tan sin sec cos tan . 434

Chapter 7 Trigonometric Identities and Equations

www.amc.glencoe.com/self_check_quiz

Find a numerical value of one trigonometric function of x.

csc x 29. 2 cot x

1 tan x 30. 2 1 cot x

1 sec x 31. cos x cot x csc x

sin x 1 cos x 32. 4 1 cos x sin x

33. cos2 x 2 sin x 2 0

34. csc x sin x tan x cos x

tan3 1 35. If sec2 1 0, find cot . tan 1 Graphing Calculator

Use a graphing calculator to determine whether each equation could be an identity. 1 1 36. 1 sin2 x cos2 x

37. cos (cos sec ) sin2

sin3 x cos3 x 38. 2 sin A (1 sin A)2 2 cos2 A 39. sin2 x cos2 x sin x cos x 40. Electronics

a. Write an expression for the power in terms of cos2 2ft.

When an alternating current of frequency f and peak current I0 passes through a resistance R, then the power delivered to the resistance at time t seconds is P I02R sin2 2ft.

Applications and Problem Solving

b. Write an expression for the power in terms of csc2 2ft.

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41. Critical Thinking

1 2

Let x tan where . Write f(x) 4x2 1

in terms of a single trigonometric function of . 42. Spherical Geometry

is the Greek letter beta and is the Greek letter gamma.

Spherical geometry is the geometry that takes place on the surface of a sphere. A line segment on the surface of the sphere is measured by the angle it subtends at the center of the sphere. Let a, b, and c be the sides of a right triangle on the surface of the sphere. Let the angles opposite those sides be , , and 90°, respectively. The following equations are true:

c a

a c b b

sin a sin sin c cos sin

cos b cos c cos a cos b. Show that cos tan a cot c. 43. Physics

When a projectile is fired from the ground, its height y and horizontal gx 2 2v0 cos

x sin cos

displacement x are related by the equation y , where v0 is 2 2 the initial velocity of the projectile, is the angle at which it was fired, and g is the acceleration due to gravity. Rewrite this equation so that tan is the only trigonometric function that appears in the equation. Lesson 7-2 Verifying Trigonometric Identities

435

Consider a circle O with radius 1. PA and T B are each perpendicular to O B . Determine the area of ABTP as a product of trigonometric functions of .

44. Critical Thinking

Let a, b, and c be the sides of a triangle. Let , , and be the respective opposite angles. Show that the area A of the triangle is given by

45. Geometry

a2 sin sin 2 sin ( )

A .

Mixed Review

tan x cos x sin x tan x 46. Simplify . (Lesson 7-1) sec x tan x. 47. Write an equation of a sine function with amplitude 2, period 180°, and phase

shift 45°. (Lesson 6-5) 15 48. Change radians to degree measure to the nearest minute. (Lesson 6-1) 16 49. Solve 3y 1 2 0. (Lesson 4-7) 3

50. Determine the equations of the vertical and horizontal asymptotes, if any, of 3x f(x) . (Lesson 3-7) x1 51. Manufacturing

The Simply Sweats Corporation makes high quality sweatpants and sweatshirts. Each garment passes through the cutting and sewing departments of the factory. The cutting and sewing departments have 100 and 180 worker-hours available each week, respectively. The fabric supplier can provide 195 yards of fabric each week. The hours of work and yards of fabric required for each garment are shown in the table below. If the profit from a sweatshirt is $5.00 and the profit from a pair of sweatpants is $4.50, how many of each should the company make for maximum profit? (Lesson 2-7)

Simply Sweats Corporation “Quality Sweatpants and Sweatshirts”

Clothing

Cutting

Sewing

Fabric

Shirt

2.5 h

1.5 yd

Pants

1.5 h

3 yd

52. State the domain and range of the relation {(16,4),(16, 4)}. Is this relation a

function? Explain. (Lesson 1-1) 53. SAT/ACT Practice A1 D 1 436

ab ba ab ba (a b)2 B (a b)2

Divide by .

Chapter 7 Trigonometric Identities and Equations

1 C a2 b2

E 0 Extra Practice See p. A38.

Have you ever had trouble p li c a ti tuning in your favorite radio station? Does the picture on your TV sometimes appear blurry? Sometimes these problems are caused by interference. Interference can result when two waves pass through the same space at the same time. The two kinds of interference are: BROADCASTING

• Use the sum and difference identities for the sine, cosine, and tangent functions.

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OBJECTIVE

Sum and Difference Identities R

7-3

• constructive interference, which occurs if the amplitude of the sum of the waves is

greater than the amplitudes of the two component waves, and • destructive interference, which occurs if the amplitude of the sum is less than the amplitudes of the component waves. What type of interference results when a signal modeled by the equation y 20 sin(3t 45°) is combined with a signal modeled by the equation y 20 sin(3t 225°)? This problem will be solved in Example 4.

Look Back You can refer to Lesson 5-8 to review the Law of Cosines.

Consider two angles and in standard position. Let the terminal side A of intersect the unit circle at point (cos , sin ) A(cos , sin ). Let the terminal side of intersect the unit circle at B(cos , sin ). We will calculate (AB)2 in two different ways.

y B

(cos , sin )

First use the Law of Cosines. (AB)2 (OA)2 (OB)2 2(OA)(OB) cos ( ) mBOA (AB)2 12 12 2(1)(1) cos ( ) OA OB 1 (AB)2 2 2 cos ( ) Simplify. Now use the distance formula. (AB)2 (cos cos )2 (sin sin )2 (AB)2 cos2 2 cos cos cos2 sin2 2 sin sin sin2 (AB)2 (cos2 sin2 ) (cos2 sin2 ) 2(cos cos sin sin ) (AB)2 1 1 2(cos cos sin sin ) cos2 a sin2 a 1 (AB)2 2 2(cos cos sin sin ) Simplify. Set the two expressions for (AB)2 equal to each other. 2 2 cos ( ) 2 2(cos cos sin sin ) 2 cos ( ) 2(cos cos sin sin ) Subtract 2 from each side. cos ( ) cos cos sin sin Divide each side by 2. This equation is known as the difference identity for cosine. Lesson 7-3

Sum and Difference Identities

437

The sum identity for cosine can be derived by substituting for in the difference identity. cos ( ) cos ( ()) cos cos () sin sin () cos cos sin (sin ) cos () cos ; sin () sin cos cos sin sin Sum and Difference Identities for the Cosine Function

If and represent the measures of two angles, then the following identities hold for all values of and . cos ( ) cos cos sin sin

Notice how the addition and subtraction symbols are related in the sum and difference identities.

You can use the sum and difference identities and the values of the trigonometric functions of common angles to find the values of trigonometric functions of other angles. Note that and can be expressed in either degrees or radians.

Example

1 a. Show by producing a counterexample that cos (x y) cos x cos y. b. Show that the sum identity for cosine is true for the values used in part a. 3

6 cos (x y) cos 3 6 cos 2

a. Let x and y . First find cos (x y) for x and y .

Replace x with 3 and y with 6. 3 6 2

0 Now find cos x cos y. 3

cos x cos y cos cos Replace x with and y with . 1 1 3 3 or 2

So, cos (x y) cos x cos y. 3

b. Show that cos (x y) cos x cos y sin x sin y for x and y .

First find cos (x y). From part a, we know that cos 0. Now find cos x cos y sin x sin y. 3

cos x cos y sin x sin y cos cos sin sin Substitute for x and y.

3 2 2 3 12 2

0 3

Thus, the sum identity for cosine is true for x and y .

438

Chapter 7

Trigonometric Identities and Equations

Example

2 Use the sum or difference identity for cosine to find the exact value of cos 735°. 735° 2(360°) 15° cos 735° cos 15°

Symmetry identity, Case 1

cos 15° cos (45° 30°) 45° and 30° are two common angles that differ by 15°. cos 45° cos 30° sin 45° sin 30° Difference identity for cosine 1 2 3 2 2

6 2 4 6 2 . Therefore, cos 735° 4

We can derive sum and difference identities for the sine function from those

These equations are examples of cofunction identities.

These equations are other cofunction identities.

for the cosine function. Replace with and with s in the identities for 2 cos ( ). The following equations result.

2 Replace s with s in the equation for cos s and with s in the 2 2 2 equation for cos s to obtain the following equations. 2 cos s sin s cos s sin s 2 2 Replace s with ( ) in the equation for cos s to derive an identity 2 for the sine of the sum of two real numbers. cos s sin s

cos s sin s

2 cos sin ( ) 2 cos cos sin sin sin 2 2 cos ( ) sin ( )

Identity for cos ( )

sin cos cos sin sin ( ) Substitute.

This equation is known as the sum identity for sine. The difference identity for sine can be derived by replacing with () in the sum identity for sine. sin [ ()] sin cos () cos sin () sin ( ) sin cos cos sin

Sum and Difference Identities for the Sine Function

If and represent the measures of two angles, then the following identities hold for all values of and . sin ( ) sin cos cos sin

Lesson 7-3

Sum and Difference Identities

439

Examples

3 Find the value of sin (x y) if 0 x 2, 0 y 2, sin x 41, and 7 25

sin y . In order to use the difference identity for sine, we need to know cos x and cos y. We can use a Pythagorean identity to determine the necessary values. sin2 cos2 1 ➡ cos2 1 sin2 Pythagorean identity Since we are given that the angles are in Quadrant I, the values of sine and cosine are positive. Therefore, cos 1 si n2 . cos x

9 1 41

1600 40 or 1681 41

cos y

7 1 25

576 24 or 625 25

Now substitute these values into the difference identity for sine. sin (x y) sin x cos y cos x sin y

491 2245 4401 275

64 1025

l Wor ea

or about 0.0624

p li c a ti

4 BROADCASTING Refer to the application at the beginning of the lesson. What type of interference results when signals modeled by the equations y 20 sin (3t 45°) and y 20 sin (3t 225°) are combined? Add the two sine functions together and simplify. 20 sin (3t 45°) 20 sin (3t 225°) 20(sin 3t cos 45° cos 3t sin 45°) 20(sin 3t cos 225° cos 3t sin 225°)

2 (cos 3t) 2 20 (sin 3t) 2

2 (cos 3t) 20(sin 3t) 2 2 2

102 sin 3t 102 cos 3t 102 sin 3t 102 cos 3t 0 The interference is destructive. The signals cancel each other completely.

You can use the sum and difference identities for the cosine and sine functions to find sum and difference identities for the tangent function. 440

Chapter 7

Trigonometric Identities and Equations

Sum and Difference Identities for the Tangent Function

If and represent the measures of two angles, then the following identities hold for all values of and . tan tan 1 tan tan

tan ( ) You will be asked to derive these identities in Exercise 47.

Example

5 Use the sum or difference identity for tangent to find the exact value of tan 285°. tan 285° tan (240° 45°) tan 240° tan 45°

1 tan 240° tan 45°

3 1

240° and 45° are common angles whose sum is 285°. Sum identity for tangent 1

3 3

Multiply by to simplify.

1 (3 )(1)

2 3

You can use sum and difference identities to verify other identities.

Example

6 Verify that csc 2 A sec A is an identity. Transform the left side since it is more complicated.

csc A sec A 1 sec A

3 sin A 2

1 sec A 3 3 sin cos A cos sin A 2 2 1 sec A (1) cos A (0) sin A 1 cos A

C HECK Communicating Mathematics

FOR

1 sin x

Reciprocal identity: csc x

Sum identity for sine 3 2

3 2

sin 1; cos 0

sec A

Simplify.

sec A sec A

Reciprocal identity

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Describe how you would convince a friend that sin (x y) sin x sin y. 2. Explain how to use the sum and difference identities to find values for the

secant, cosecant, and cotangent functions of a sum or difference. Lesson 7-3 Sum and Difference Identities

441

3. Write an interpretation of the identity sin (90° A) cos A in

terms of a right triangle.

4. Derive a formula for cot ( ) in terms of cot and cot . Guided Practice

Use sum or difference identities to find the exact value of each trigonometric function. 5. cos 165°

6. tan 12

7. sec 795°

Find each exact value if 0 x and 0 y . 4 1 8. sin (x y) if sin x and sin y 9 4 5 5 9. tan (x y) if csc x and cos y 13 3

Verify that each equation is an identity.

11. tan cot 2

10. sin (90° A) cos A 1 cot x tan y 12. sin (x y) csc x sec y

13. Electrical Engineering

is the Greek letter omega.

Analysis of the voltage in certain types of circuits involves terms of the form sin (n0t 90°), where n is a positive integer, 0 is the frequency of the voltage, and t is time. Use an identity to simplify this expression.

E XERCISES Practice

Use sum or difference identities to find the exact value of each trigonometric function.

442

14. cos 105°

15. sin 165°

7 16. cos 12

17. sin 12

18. tan 195°

19. cos 12

20. tan 165°

23 21. tan 12

22. sin 735°

23. sec 1275°

5 24. csc 12

113 25. cot 12

Chapter 7 Trigonometric Identities and Equations

www.amc.glencoe.com/self_check_quiz

Find each exact value if 0 x and 0 y . 8 12 26. sin (x y) if cos x and sin y 17 37 3 4 27. cos (x y) if cos x and cos y 5 5 8 3 28. tan (x y) if sin x and cos y 17 5 5 1 29. cos (x y) if tan x and sin y 3 3 6 3 30. tan (x y) if cot x and sec y 5 2 5 12 31. sec (x y) if csc x and tan y 3 5 1 2 32. If and are two angles in Quadrant I such that sin and cos , find 5 7

sin ( ).

1 3 33. If x and y are acute angles such that cos x and cos y , what is the value 3 4

of cos (x y)?

Verify that each equation is an identity.

34. cos x sin x 2

35. cos (60° A) sin (30° A)

36. sin (A ) sin A

37. cos (180° x) cos x

1 tan x 38. tan (x 45°) 1 tan x

tan A tan B 39. sin (A B) sec A sec B

1 tan A tan B 40. cos (A B) sec A sec B

sec A sec B 41. sec (A B) 1 tan A tan B

42. sin (x y) sin (x y) sin2 x sin2 y

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43. Electronics

In an electric circuit containing a capacitor, inductor, and resistor the voltage drop across the inductor is given by VL I0L cos t , where 2 I0 is the peak current, is the frequency, L is the inductance, and t is time. Use the sum identity for cosine to express VL as a function of sin t.

Applications and Problem Solving

p li c a ti

44. Optics

The index of refraction for a medium through which light is passing is the ratio of the velocity of light in free space to the velocity of light in the medium. For light passing symmetrically through a glass prism, the index of refraction n is given by the equation n

1 sin ( ) 2 sin 2

, where is the

deviation angle and is the angle of the apex of the prism as shown in the 2

diagram. If 60°, show that n 3 sin cos . Lesson 7-3 Sum and Difference Identities

443

45. Critical Thinking

Simplify the following expression without expanding any of the sums or differences.

sin A cos A cos A sin A

f(x h) f(x) h

In calculus, you will explore the difference quotient .

46. Calculus

a. Let f(x) sin x. Write and expand an expression for the difference quotient. b. Set your answer from part a equal to y. Let h 0.1 and graph. c. What function has a graph similar to the graph in part b? 47. Critical Thinking

Derive the sum and difference identities for the tangent

function. 48. Critical Thinking

Consider the following theorem. If A, B, and C are the angles of a nonright triangle, then tan A tan B tan C tan A tan B tan C.

a. Choose values for A, B, and C. Verify that the conclusion is true for your

specific values. b. Prove the theorem.

Mixed Review

1 cos2 x 49. Verify the identity sec2 x csc2 x cot2 x. (Lesson 7-2) 1 sin2 x 1 3 50. If sin and , find tan . (Lesson 7-1) 8 2

51. Find sin Arctan 3 . (Lesson 6-8) 52. Find the values of for which csc is undefined. (Lesson 6-7) 53. Weather

The average seasonal high temperatures for Greensboro, North Carolina, are given in the table. Write a sinusoidal function that models the temperatures, using t 1 to represent winter. (Lesson 6-6) Winter

Spring

Summer

Fall

50°

70°

86°

71°

Source: Rand McNally & Company

54. State the amplitude, period, and phase shift for the function y 8 cos ( 30°).

(Lesson 6-5) 55. Find the value of sin (540°). (Lesson 6-3) 56. Geometry

A sector has arc length of 18 feet and a central angle measuring 2.9 radians. Find the radius and the area of the sector. (Lesson 6-1)

57. Navigation

A ship at sea is 70 miles from one radio transmitter and 130 miles from another. The angle formed by the rays from the ship to the transmitters measures 130°. How far apart are the transmitters? (Lesson 5-8)

58. Determine the number of possible solutions for a triangle if A 120°, b 12,

and a 4. (Lesson 5-7)

444

Chapter 7 Trigonometric Identities and Equations

59. Photography

A photographer observes a 35-foot totem pole that stands vertically on a uniformly-sloped hillside and the shadow cast by it at different times of day. At a time when the angle of elevation of the sun is 37°12, the shadow of the pole extends directly down the slope. This is the effect that the photographer is seeking. If the hillside has an angle of inclination of 6°40, find the length of the shadow. (Lesson 5-6)

60. Find the roots of the equation

4x3 3x2 x 0. (Lesson 4-1) 61. Solve x 1 4. (Lesson 3-3) 62. Find the value of the determinant

(Lesson 2-5)



1 2 . 3 6

63. If f(x) 3x 2 4 and g(x) 5x 1, find f g(4)

and g f(4). (Lesson 1-2)

64. SAT Practice

What is the value of

(8)62 862? A 1 B 0 C 1 D 8 E 62

MID-CHAPTER QUIZ Use the given information to determine the exact trigonometric value. (Lesson 7-1) 2 1. sin , 0 ; cot 7 2 4 2. tan , 90° 180°; cos 3 19 3. Express cos as a trigonometric 4

function of an angle in Quadrant I.

Verify that each equation is an identity. (Lessons 7-2 and 7-3) 6. cot x sec x sin x 2 tan x cos x csc x 1 cot tan 7. tan ( ) cot tan 8. Use a sum or difference identity to find the exact value of cos 75°. (Lesson 7-3)

(Lesson 7-1)

Verify that each equation is an identity. (Lesson 7-2) 1 1 4. 1 1 tan2 x 1 cot2 x csc2 sec2 5. csc2 sec2

Extra Practice See p. A38.

Find each exact value if 0 x and 2 0 y . (Lesson 7-3) 2

2 3 9. cos (x y) if sin x and sin y 3 4 5 10. tan (x y) if tan x and sec y 2 4

Lesson 7-3 Sum and Difference Identities

445

GRAPHING CALCULATOR EXPLORATION

7-3B Reduction Identities OBJECTIVE • Identify equivalent values for trigonometric functions involving quadrantal angles.

Example

You may recall from Chapter 6 that a phase shift of 90° right for the cosine function results in the sine function.

An Extension of Lesson 7-3 In Chapter 5, you learned that any trigonometric function of an acute angle is equal to the cofunction of the complement of the angle. For example, sin cos (90° ). This is a part of a large family of identities called the reduction identities. These identities involve adding and subtracting the quandrantal angles, 90°, 180°, and 270°, from the angle measure to find equivalent values of the trigonometric function. You can use your knowledge of phase shifts and reflections to find the components of these identities.

Find the values of the sine and cosine functions for 90°, 180°, and 270° that are equivalent to sin . 90° Graph y sin , y sin ( 90°), and y cos ( 90°), letting X in degree mode represent . You can select different display formats to help you distinguish the three graphs. Note that the graph of y cos (X 90) is the same as the graph of y sin X. This suggests that sin cos ( 90°). Remember that an identity must be proved algebraically. A graph does not prove an identity. 180° Graph y sin , y sin ( 180°), and y cos ( 180°) using X to represent . Discount y cos ( 180°) as a possible equivalence because it would involve a phase shift, which would change the actual value of the angle being considered.

Y1 sin (X) Y3 cos (X90)

Y2 sin (X90)

[0, 360] scl:90 by [2, 2] scl:1

Y3 cos (X180)

Y2 sin (X180)

Y1 sin (X) [0, 360] scl:90 by [2, 2] scl:1

Note that the graph of sin ( 180°) is a mirror reflection of sin . Remember that a reflection over the x-axis results in the mapping (x, y) → (x, y). So to obtain a graph that is identical to y sin , we need the reflection of y sin ( 180°) over the x-axis, or y sin ( 180°). Thus, sin sin ( 180°). Graph the two equations to investigate this equality. 446 Chapter 7 Trigonometric Identities and Equations

270° In this case, sin ( 270°) is a phase shift, so ignore it. The graph of cos ( 270°) is a reflection of sin over the x-axis. So, sin cos ( 270°).

Y3 cos (X270)

Y1 sin (X)

Y2 sin (X270) [0, 360] scl:90 by [2, 2] scl:1

The family of reduction identities also contains the relationships among the other cofunctions of tangent and cotangent and secant and cosecant. In addition to 90°, 180°, and 270° angle measures, the reduction identities address other measures such as 90° , 180° , 270° , and 360° .

TRY THESE

WHAT DO YOU THINK?

Copy and complete each statement with the proper trigonometric functions. 1. cos

( 90°)

( 180°)

( 270°)

2. tan

( 90°)

( 180°)

( 270°)

3. cot

( 90°)

( 180°)

( 270°)

4. sec

( 90°)

( 180°)

( 270°)

5. csc

( 90°)

( 180°)

( 270°)

6. Suppose the expressions involving subtraction in Exercises 1-5 were changed to sums. a. Copy and complete each statement with the proper trigonometric functions. (1) sin ? ( 90°) ? ( 180°) ? ( 270°) (2) cos ? ( 90°) ? ( 180°) ? ( 270°) (3) tan ? ( 90°) ? ( 180°) ? ( 270°) (4) cot ? ( 90°) ? ( 180°) ? ( 270°) (5) sec ? ( 90°) ? ( 180°) ? ( 270°) (6) csc ? ( 90°) ? ( 180°) ? ( 270°) b. How do the identities in part a compare to those in Exercises 1-5? 7. a. Copy and complete each statement with the proper trigonometric functions. (1) sin ? (90° ) ? (180° ) ? (270° ) (2) cos ? (90° ) ? (180° ) ? (270° ) (3) tan ? (90° ) ? (180° ) ? (270° ) (4) cot ? (90° ) ? (180° ) ? (270° ) (5) sec ? (90° ) ? (180° ) ? (270° ) (6) csc ? (90° ) ? (180° ) ? (270° ) b. How do the identities in part a compare to those in Exercise 6a? 8. a. How did reduction identities get their name? b. If you needed one of these identities, but could not remember it, what other type(s) of identities could you use to derive it? Lesson 7-3B Reduction Identities

447

7-4 Double-Angle and Half-Angle Identities ARCHITECTURE

Mike MacDonald is an architect p li c a ti who designs water fountains. One part of his job is determining the placement of the water jets that shoot the water into the air to create arcs. These arcs are modeled by parabolic functions. When a stream of water is shot into the air with velocity v at an angle of with the horizontal, the model predicts that the water will travel a Ap

• Use the doubleand half-angle identities for the sine, cosine, and tangent functions.

l Wor ea

OBJECTIVE

v2 g

horizontal distance of D sin 2 and v2

reach a maximum height of H sin2 , 2g where g is the acceleration due to gravity. The ratio of H to D helps determine the total height H

and width of the fountain. Express as a function D of . This problem will be solved in Example 3.

It is sometimes useful to have identities to find the value of a function of twice an angle or half an angle. We can substitute for both and in sin ( ) to find an identity for sin 2. sin 2 sin ( ) sin cos cos sin

Sum identity for sine

2 sin cos The same method can be used to find an identity for cos 2. cos 2 cos ( ) cos 2 cos cos sin sin

Sum identity for cosine

cos2 sin2 If we substitute 1 cos2 for sin2 or 1 sin2 for cos2 , we will have two alternate identities for cos 2. cos 2 2 cos2 1 cos 2 1 2 sin2 These identities may be used if is measured in degrees or radians. So, may represent either a degree measure or a real number. 448

Chapter 7

Trigonometric Identities and Equations

The tangent of a double angle can be found by substituting for both and in tan ( ). tan 2 tan ( ) tan tan 1 tan tan

Sum identity for tangent

2 tan 1 tan

If represents the measure of an angle, then the following identities hold for all values of . sin 2 2 sin cos cos 2 cos2 sin2 cos 2 2 cos2 1 cos 2 1 2 sin2

Double-Angle Identities

2 tan tan 2 1 tan2

Example

1 If sin 3 and has its terminal side in the first quadrant, find the exact value of each function. a. sin 2 To use the double-angle identity for sin 2, we must first find cos . sin2 cos2 1

2 3

cos2 1

sin

5 9

cos2

5 cos 3

Then find sin 2. sin 2 2 sin cos

5 23 3

2 5 sin ; cos 3

45 9

b. cos 2 Since we know the values of cos and sin , we can use any of the doubleangle identities for cosine. cos 2 cos2 sin2

5 3

23 2

5 ; sin 2 cos 3

1 9 Lesson 7-4

Double-Angle and Half-Angle Identities

449

c. tan 2 We must find tan to use the double-angle identity for tan 2. sin cos 2 3

tan

sin , cos 2 3

5 3

2 25 or

Then find tan 2. 2 tan 1 tan

tan 2 2

25 2 5

25 1 5

25 tan 5

4 5 5 or 45 1 5

d. cos 4 Since 4 2(2), use a double-angle identity for cosine again. cos 4 cos 2(2) cos2 (2) sin2 (2) Double-angle identity 45 19 9

1 45 (parts a and b) cos 2 , sin 2 9

79 81

We can solve two of the forms of the identity for cos 2 for cos and sin , respectively, and the following equations result.

450

Chapter 7

1 cos 2 2

cos 2 2 cos2 1

Solve for cos .

cos

cos 2 1 2 sin2

Solve for sin .

sin

Trigonometric Identities and Equations

1 cos 2 2

We can replace 2 with and with to derive the half-angle identities. sin 2 tan 2 cos 2 1 cos 2 1 cos 2

1 co s

1 cos or

If represents the measure of an angle, then the following identities hold for all values of . 1 cos 2 1 cos cos 2 2 2

sin

Half-Angle Identities

1 cos 1 c os

tan , cos 1 Unlike with the double-angles identities, you must determine the sign.

Example

2 Use a half-angle identity to find the exact value of each function. 7 12

a. sin 7

7 6 sin sin 12 2

1 cos 6 2

1 cos 2

7 12

Use sin . Since is in Quadrant II, choose the positive sine value.

3 1 2 2

3 2

b. cos 67.5° 135° 2

cos 67.5° cos

1 cos 135° 2

1 cos 2

Use cos . Since 67.5° is in Quadrant I, choose the positive cosine value.

2 1 2 2

2 2 2

Lesson 7-4

Double-Angle and Half-Angle Identities

451

Double- and half-angle identities can be used to simplify trigonometric expressions.

H D

a. Find and simplify .

l Wor ea

3 ARCHITECTURE Refer to the application at the beginning of the lesson.

Example

b. What is the ratio of the maximum height of the water to the horizontal distance it travels for an angle of 27°?

p li c a ti

v2 sin2 2g H a. D v2 sin 2 g sin2 2 sin 2 sin2 4 sin cos 1 sin 4 cos 1 tan 4

Simplify. sin 2 2 sin cos Simplify. sin cos

Quotient identity: tan

Therefore, the ratio of the maximum height of the water to the horizontal 1 4

distance it travels is tan . H D

1 4

b. When 27°, tan 27°, or about 0.13. For an angle of 27°, the ratio of the maximum height of the water to the horizontal distance it travels is about 0.13.

The double- and half-angle identities can also be used to verify other identities.

Example

cos 2

cot 1

is an identity. 4 Verify that 1 sin 2 cot 1 cos 2 cot 1 1 sin 2 cot 1 cos 1 sin cos 2 cos Reciprocal identity: cot cos 1 sin 2 sin 1 sin

452

Chapter 7

cos 2 cos sin cos sin 1 sin 2

Multiply numerator and denominator by sin .

cos 2 cos sin cos sin 1 sin 2 cos sin cos sin

Multiply each side by 1.

cos2 sin2 cos 2 cos2 2 cos sin sin2 1 sin 2

Multiply.

Trigonometric Identities and Equations

cos2 sin2 cos 2 1 2 cos sin 1 sin 2

Simplify.

cos 2 cos 2 1 sin 2 1 sin 2

Double-angle identities: cos 2 sin 2 cos 2, 2 cos sin sin 2

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Write a paragraph about the conditions under which you would use each of the

three identities for cos 2.

1 cos 2. Derive the identity sin from cos 2 1 2 sin2 . 2 2 3. Name the quadrant in which the terminal side lies. a. x is a second quadrant angle. In which quadrant does 2x lie? x b. is a first quadrant angle. In which quadrant does x lie? 2 x c. 2x is a second quadrant angle. In which quadrant does lie? 2 4. Provide a counterexample to show that sin 2 2 sin is not an identity. 5. You Decide

Tamika calculated the exact value of sin 15° in two different ways.

6 2 . When she used the Using the difference identity for sine, sin 15° was 4

3 2 half-angle identity, sin 15° equaled . Which answer is correct? 2

Explain. Guided Practice

Use a half-angle identity to find the exact value of each function. 6. sin 8

7. tan 165°

Use the given information to find sin 2, cos 2, and tan 2. 4 3 9. tan , 3 2

2 8. sin , 0° 90° 5

Verify that each equation is an identity. 2 10. tan 2 cot tan 1 sec A sin A 11. 1 sin 2A 2 sec A x x sin x 12. sin cos 2 2 2 13. Electronics

Consider an AC circuit consisting of a power supply and a resistor. If the current in the circuit at time t is I0 sin t, then the power delivered to the resistor is P I02 R sin2 t, where R is the resistance. Express the power in terms of cos 2t.

www.amc.glencoe.com/self_check_quiz

Lesson 7-4 Double-Angle and Half-Angle Identities

453

E XERCISES Practice

Use a half-angle identity to find the exact value of each function.

14. cos 15°

15. sin 75°

5 16. tan 12

3 17. sin 8

7 18. cos 12

19. tan 22.5°

1 20. If is an angle in the first quadrant and cos , find tan . 2 4

Use the given information to find sin 2, cos 2, and tan 2.

4 21. cos , 0° 90° 5

1 22. sin , 0 3 2

23. tan 2, 2

4 24. sec , 90° 180° 3

3 25. cot , 180° 270° 2

5 3 26. csc , 2 2 2

2 , find tan 2. 27. If is an angle in the second quadrant and cos 3 Verify that each equation is an identity.

l Wor ea

Applications and Problem Solving

p li c a ti

1 28. csc 2 sec csc 2

cos 2A 29. cos A sin A cos A sin A

30. (sin cos )2 1 sin 2

cos 2x 1 31. cos x 1 2(cos x 1)

cos2 sin2 32. sec 2 cos2 sin2

sin A A 33. tan 1 cos A 2

34. sin 3x 3 sin x 4 sin3 x

35. cos 3x 4 cos3 x 3 cos x

36. Architecture

Refer to the application at the beginning of the lesson. If the angle of the water is doubled, what is the ratio of the new maximum height to the original maximum height?

37. Critical Thinking

Circle O is a unit circle. Use the figure

sin 1 to prove that tan . 1 cos 2

B P O A D

Suppose a projectile is launched with velocity v at an angle to the horizontal from the base of a hill that makes an angle with the horizontal ( ). Then the range of the projectile, measured along the slope

38. Physics

2v2 cos sin ( )

of the hill, is given by R . Show that if 45°, then g cos2 v22 (sin 2 cos 2 1). R g

454

Chapter 7 Trigonometric Identities and Equations

39. Geography

The Mercator projection of the globe is a projection on which the distance between the lines of latitude increases with their distance from the equator. The calculation of the location of a point on this projection involves the

Research For the latitude and longitude of world cities, and the distance between them, visit: www.amc. glencoe.com

expression tan 45° , 2 where L is the latitude of the point. a. Write this expression in

terms of a trigonometric function of L. b. Find the value of this expression if L 60°. 40. Critical Thinking

Determine the tangent of angle

in the figure.

30˚ 21

Mixed Review

41. Find the exact value of sec . (Lesson 7-3) 12 42. Show that sin x2 cos x2 1 is not an identity. (Lesson 7-1) 43. Find the degree measure to the nearest tenth of the central angle of a circle of

radius 10 centimeters if the measure of the subtended arc is 17 centimeters. (Lesson 6-1) 44. Surveying

To find the height of a mountain peak, points A and B were located on a plain in line with the peak, and the angle of elevation was measured from each point. The angle at A was 36°40 , and the angle at B was 21°10 . The distance from A to B was 570 feet. How high is the peak above the level of the plain? (Lesson 5-4)

21˚10'

36˚ 40'

A 570 ft

45. Write a polynomial equation of least degree with roots 3, 0.5, 6, and 2.

(Lesson 4-1) 46. Graph y 2x 5 and its inverse. (Lesson 3-4) 47. Solve the system of equations. (Lesson 2-1)

x 2y 11 3x 5y 11 48. SAT Practice Extra Practice See p. A39.

Grid-In If (a b)2 64, and ab 3, find a2 b2. Lesson 7-4 Double-Angle and Half-Angle Identities

455

7-5 Solving Trigonometric Equations R

• Solve trigonometric equations and inequalities.

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ENTERTAINMENT

OBJECTIVE

When you ride a Ferris wheel that has a diameter of 40 meters and turns at a rate of 1.5 revolutions per minute, the height above the ground, in meters, of your seat after t minutes can be modeled by the equation h 21 20 cos 3t. How long after the ride starts will your seat first be 31 meters above the ground? This problem will be solved in Example 4. p li c a ti

So far in this chapter, we have studied a special type of trigonometric equation called an identity. Trigonometric identities are equations that are true for all values of the variable for which both sides are defined. In this lesson, we will examine another type of trigonometric equation. These equations are true for only certain values of the variable. Solving these equations resembles solving algebraic equations. Most trigonometric equations have more than one solution. If the variable is not restricted, the periodic nature of trigonometric functions will result in an infinite number of solutions. Also, many trigonometric expressions will take on a given value twice every period. If the variable is restricted to two adjacent quadrants, a trigonometric equation will have fewer solutions. These solutions are called principal values. For sin x and tan x, the principal values are in Quadrants I and IV. So x is in the interval 90° x 90°. For cos x, the principal values are in Quadrants I and II, so x is in the interval 0° x 180°.

Example

1 Solve sin x cos x 2 cos x 0 for principal values of x. Express solutions in degrees. sin x cos x 1 cos x 0 2

cos x sin x 1 0 Factor. cos x 0 x 90°

sin x 1 0 2

sin x 1 2

x 30° The principal values are 30° and 90°.

456

Chapter 7

Trigonometric Identities and Equations

Set each factor equal to 0.

If an equation cannot be solved easily by factoring, try writing the expressions in terms of only one trigonometric function. Remember to use your knowledge of identities.

Example

2 Solve cos2 x cos x 1 sin2 x for 0 x 2. This equation can be written in terms of cos x only. cos2 x cos x 1 sin2 x cos2 x cos x 1 1 cos2 x Pythagorean identity: sin2 x 1 cos2 x 2 cos2 x cos x 0 cos x(2 cos x 1) 0

Factor.

cos x 0

2 cos x 1 0

3 x or x 2

cos x 1

2 5 x or x 3 3

3 5 , and . The solutions are , , 3 2

As indicated earlier, most trigonometric equations have infinitely many solutions. When all of the values of x are required, the solution should be represented as x 360k° or x 2k for sin x and cos x and x 180k° or x k for tan x, where k is any integer.

Example

3 Solve 2 sec2 x tan4 x 1 for all real values of x. A Pythagorean identity can be used to write this equation in terms of tan x only. 2 sec2 x tan4 x 1 2(1 tan2 x) tan4 x 1 2 2 tan2 x tan4 x 1

Pythagorean identity: sec2 x 1 tan2 x Simplify.

tan4 x 2 tan2 x 3 0 (tan2 x 3)(tan2 x 1) 0 tan2 x 3 0

Factor. tan2 x 1 0

tan2 x 3 When a problem asks for real values of x, use radians.

tan2 x 1

This part gives no solutions since tan x 3 tan2 x 0. x k or x k, where k is any integer. 3

3 The solutions are k and k. 3 3

There are times when a general expression for all of the solutions is helpful for determining a specific solution. Lesson 7-5

Solving Trigonometric Equations

457

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Example

p li c a ti

4 ENTERTAINMENT Refer to the application at the beginning of the lesson. How long after the Ferris wheel starts will your seat first be 31 meters above the ground? h 21 20 cos 3t 31 21 20 cos 3t Replace h with 31. 1 2 2 2k 3t 3 2 2 k t 9 3

cos 3t

4 2k 3t 3 4 2 k t 9 3

or or

where k is any integer

The least positive value for t is obtained by letting k 0 in the first expression. 2 Therefore, t of a minute or about 13 seconds. 9

You can solve some trigonometric inequalities using the same techniques as for algebraic inequalities. The unit circle can be useful when deciding which angles to include in the answer.

Example

5 Solve 2 sin 1 0 for 0 2. 2 sin 1 0 1 2

sin

Solve for sin .

In terms of the unit circle, we need to find 1 points with y-coordinates greater than .

2 1 7 The values of for which sin are 2 6 11 and . The figure shows that the solution of 6 7 11 the inequality is 0 or 2. 6 6

(

3 , 1 2 2

)

(

3, 1 2 2

GRAPHING CALCULATOR EXPLORATION Some trigonometric equations and inequalities are difficult or impossible to solve with only algebraic methods. A graphing calculator is helpful in such cases.

TRY THESE Graph each side of the

4. What do the values in Exercise 3 represent? How could you verify this conjecture? 5. Graph y 2 cos x sin x for 0 x 2.

2. tan 0.5x cos x for 2 x 2

a. How could you use the graph to solve the equation sin x 2 cos x? How does this solution compare with those found in Exercise 3?

3. Use the CALC menu to find the intersection point(s) of the graphs in Exercises 1 and 2.

b. What equation would you use to apply this method to tan 0.5x cos x?

equation as a separate function. 1. sin x 2 cos x for 0 x 2

458

WHAT DO YOU THINK?

Chapter 7 Trigonometric Identities and Equations

)

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Explain the difference between a trigonometric identity and a trigonometric

equation that is not an identity. 2. Explain why many trigonometric equations have infinitely many solutions. 3. Write all the solutions to a trigonometric equation in terms of sin x, given that

the solutions between 0° and 360° are 45° and 135°. 4. Math

Journal Compare and contrast solving trigonometric equations with solving linear and quadratic equations. What techniques are the same? What techniques are different? How many solutions do you expect? Do you use a graphing calculator in a similar manner?

Guided Practice

Solve each equation for principal values of x. Express solutions in degrees. 6. 2 cos x 3 0

5. 2 sin x 1 0

Solve each equation for 0° x 360°.

3 7. sin x cot x 2

8. cos 2x sin2 x 2

Solve each equation for 0 x 2. 9. 3 tan2 x 1 0

10. 2 sin2 x 5 sin x 3

Solve each equation for all real values of x. 11. sin2 2x cos2 x 0

12. tan2 x 2 tan x 1 0

13. cos2 x 3 cos x 2

14. sin 2x cos x 0

15. Solve 2 cos 1 0 for 0 2. 16. Physics

The work done in moving an object through a displacement of d meters is given by W Fd cos , where is the angle between the displacement and the force F exerted. If Lisa does 1500 joules of work while exerting a 100-newton force over 20 meters, at what angle was she exerting the force?

E XERCISES Practice

Solve each equation for principal values of x. Express solutions in degrees.

17. 2 sin x 1 0

18. 2 cos x 1 0

19. sin 2x 1 0

20. tan 2x 3 0

21. cos2 x cos x

22. sin x 1 cos2 x

Solve each equation for 0° x 360°.

23. 2 cos x 1 0

1 24. cos x tan x 2

25. sin x tan x sin x 0

26. 2 cos2 x 3 cos x 2 0

27. sin 2x sin x

28. cos (x 45°) cos (x 45°) 2

www.amc.glencoe.com/self_check_quiz

Lesson 7-5 Solving Trigonometric Equations

459

sin 0 in the interval 0° 360°. 29. Find all solutions to 2 sin cos 3 Solve each equation for 0 x 2. 30. (2 sin x 1)(2 cos2 x 1) 0

31. 4 sin2 x 1 4 sin x

32. 2 tan x 2 sin x

33. sin x cos 2x 1

34. cot2 x csc x 1

35. sin x cos x 0

36. Find all values of between 0 and 2 that satisfy 1 3 sin cos 2.

Solve each equation for all real values of x.

1 37. sin x 2

38. cos x tan x 2 cos2 x 1

39. 3 tan2 x 3 tan x

40. 2 cos2 x 3 sin x

1 41. cos x sin x cos x sin x

42. 2 tan2 x 3 sec x 0

1 43. sin x cos x 2

3 44. cos2 x sin2 x 2

45. sin4 x 1 0

46. sec2 x 2 sec x 0

47. sin x cos x 1

48. 2 sin x csc x 3

Solve each inequality for 0 2.

3 49. cos 2

1 50. cos 0 2

51.

Solve each equation graphically on the interval 0 x 2.

Applications and Problem Solving

55. Optics

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Graphing Calculator

p li c a ti

x 53. sin x 0 2

52. tan x 0.5

When light passes through a small slit, it is diffracted. The angle subtended by the first diffraction minimum is related to the wavelength of the light and the width D of the slit by the equation sin . Consider light of wavelength D 550 nanometers (5.5 107 m). What is the angle subtended by the first diffraction minimum when the light passes through a slit of width 3 millimeters?

56. Critical Thinking

2 sin 1 0

54. cos x 3 sin x

Intensity of light

Solve the inequality sin 2x sin x for 0 x 2 without a

calculator. 57. Physics

The range of a projectile that is launched with an initial velocity v at v2 an angle of with the horizontal is given by R sin 2, where g is the g acceleration due to gravity or 9.8 meters per second squared. If a projectile is launched with an initial velocity of 15 meters per second, what angle is required to achieve a range of 20 meters?

460

Chapter 7 Trigonometric Identities and Equations

58. Gemology

The sparkle of a diamond is created by refracted light. Light travels at different speeds in different mediums. When light rays pass from one medium to another in which they travel at a different velocity, the light is bent, or refracted. According to Snell’s Law, n1 sin i n2 sin r, where n1 is the index of refraction of the medium the light is exiting, n2 is the index of refraction of the medium the light is entering, i is the angle of incidence, and r is the angle of refraction. a. The index of refraction of a diamond is 2.42, and the index of

refraction of air is 1.00. If a beam of light strikes a diamond at an angle of 35°, what is the angle of refraction? b. Explain how a gemologist might use Snell’s Law to determine if a

diamond is genuine. 59. Music

A wave traveling in a guitar string can be modeled by the equation D 0.5 sin(6.5x) sin(2500t), where D is the displacement in millimeters at the position x meters from the left end of the string at time t seconds. Find the first positive time when the point 0.5 meter from the left end has a displacement of 0.01 millimeter.

D x

How many solutions in the interval 0° x 360° should a you expect for the equation a sin(bx c) d d , if a 0 and b is a 2 positive integer?

60. Critical Thinking

The point P(x, y) can be rotated degrees counterclockwise x about the origin by multiplying the matrix on the left by the rotation matrix y cos sin R . Determine the angle required to rotate the point P(3, 4) to sin cos

61. Geometry

the point P(17 , 22). Mixed Review

62. Find the exact value of cot 67.5°. (Lesson 7-4)

tan x 2 63. Find a numerical value of one trigonometric function of x if . sec x 5

(Lesson 7-2)

2 64. Graph y cos . (Lesson 6-4) 3 65. Transportation A boat trailer has wheels with a diameter of 14 inches. If the

trailer is being pulled by a car going 45 miles per hour, what is the angular velocity of the wheels in revolutions per second? (Lesson 6-2) 66. Use the unit circle to find the value of csc 180°. (Lesson 5-3) 67. Determine the binomial factors of x3 3x 2. (Lesson 4-3) 68. Graph y x3 3x 5. Find and classify its extrema. (Lesson 3-6) 69. Find the values of x and y for which

is true. (Lesson 2-3) 3x 46 16 2y

70. Solve the system x y z 1, 2x y 3z 5, and x y z 11. (Lesson 2-2) 71. Graph g(x) x 3. (Lesson 1-7)

If AC 6, what is the area of triangle ABC?

72. SAT/ACT Practice A1 D6 Extra Practice See p. A39.

B 6 E 12

B 1

C 3

A Lesson 7-5 Solving Trigonometric Equations

C 461

MATHEMATICS TRIGONOMETRY Trigonometry was developed in response to the needs of astronomers. In fact, it was not until the thirteenth century that trigonometry and astronomy were treated as separate disciplines. Early Evidence

The earliest use of trigonometry was to calculate the heights of objects using the lengths of shadows. Egyptian mathematicians produced tables relating the lengths of shadows at particular times of day as early as the thirteenth century B.C.

In about 100 A.D., the Greek mathematician Menelaus, credited with the first work on spherical trigonometry, also produced a treatise on chords in a circle. Ptolemy, a Babylonian mathematician, produced yet another book of chords, believed to have been adapted from Hipparchus’ treatise. He used an identity similar to sin2 x cos2 x 1, except that it was relative to chords. He also used the formulas sin (x y) sin x cos y a b c cos x sin y and as they sin A sin B sin C related to chords. In about 500 A.D., Aryabhata, a Hindu mathematician, was the first person to use the sine function as we know it today. He produced a table of sines and called the sine jya. In 628 A.D., another Hindu mathematician, Brahmagupta, also produced a table of sines.

Chapter 7 Trigonometric Identities and Equations

Many mathematicians developed theories and applications of trigonometry during this time period. Nicolas Copernicus (1473–1543) published a book highlighting all the trigonometry necessary for astronomy at that time. During this period, the sine and versed sine were the most important trigonometric functions. Today, the versed sine, which is defined as versin x 1 cos x, is rarely used.

Copernicus

The Greek mathematician Hipparchus (190–120 B.C.), is generally credited with laying the foundations of trigonometry. Hipparchus is believed to have produced a twelve-book treatise on the construction of a table of chords. This table related the A lengths of chords of a circle to the angles subtended by those chords. In the O diagram, AOB would be B compared to the length of chord AB .

462

The Renaissance

Modern Era Mathematicians of the 1700s, 1800s, and 1900s worked on more sophisticated trigonometric ideas such as those relating to complex variables and hyperbolic functions. Renowned mathematicians who made contributions to trigonometry during this era were Bernoulli, Cotes, DeMoivre, Euler, and Lambert.

Today architects, such as Dennis Holloway of Santa Fe, New Mexico, use trigonometry in their daily work. Mr. Holloway is particularly interested in Native American designs. He uses trigonometry to determine the best angles for the walls of his buildings and for finding the correct slopes for landscaping.

ACTIVITIES 1. Draw a circle of radius 5 centimeters.

Make a table of chords for angles of measure 10° through 90°(use 10° intervals). The table headings should be “angle measure” and “length of chord.” (In the diagram of circle O, you are using AOB and chord AB .) 2. Find out more about personalities

referenced in this article and others who contributed to the history of trigonometry. Visit www.amc.glencoe.com •

When a discus thrower releases the p li c a ti discus, it travels in a path that is tangent to the circle traced by the discus while the thrower was spinning around. Suppose the center of motion is the origin in the coordinate system. If the thrower spins so that the discus traces the unit circle and the discus is released at (0.96, 0.28), find an equation of the line followed by the discus. This problem will be solved in Example 2. TRACK AND FIELD

• Write the standard form of a linear equation given the length of the normal and the angle it makes with the x-axis. • Write linear equations in normal form.

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OBJECTIVES

Normal Form of a Linear Equation R

7-6

You are familiar with several forms for the equation of a line. These include slope-intercept form, point-slope form, and standard form. The usefulness of each form in a particular situation depends on how much relevant information each form provides upon inspection. In this lesson, you will learn about the normal form of a linear equation. The normal form uses trigonometry to provide information about the line.

is the Greek letter phi.

In general, a normal line is a line that is perpendicular to another line, curve, or surface. Given a line in the xy-plane, there is a normal line that intersects the given line and passes through the origin. The angle between this normal line and the x-axis is denoted by . The normal form of the equation of the given line is written in terms of and the length p of the segment of the normal line from the given line to the origin. Suppose is a line that does not pass through the origin and p is the length of the normal from the origin. Let C be the point of intersection of with the normal, and let be the positive angle formed by the x-axis and MC OC . Draw perpendicular to the x-axis. OM Since is in standard position, cos or MC p

C (p cos , p sin )

OM p cos and sin or MC p sin . p sin p cos

sin cos

So or is the slope of OC . Since is perpendicular to O C , the slope of is the negative reciprocal of the slope of O C , or cos . sin

Lesson 7-6

A M

Normal Form of a Linear Equation

463

Look Back You can refer to Lesson 1-4 to review the point-slope form.

Since contains C, we can use the point-slope form to write an equation of line . y y1 m(x x1) cos sin

y p sin (x p cos ) Substitute for m, x1, and y1. y sin p sin2 x cos p cos2 x cos y sin

p(sin2

cos2

Multiply each side by sin .

)

x cos y sin p 0

sin2 cos2 1

The normal form of a linear equation is Normal Form

x cos y sin p 0, where p is the length of the normal from the line to the origin and is the positive angle formed by the positive x-axis and the normal.

You can write the standard form of a linear equation if you are given the values of and p.

Examples 1 Write the standard form of the equation of a line for which the length of the normal segment to the origin is 6 and the normal makes an angle of 150° with the positive x-axis. x cos y sin p 0 x cos 150° y sin 150° 6 0 1 3 x y 6 0 2

Normal form 150° and p 6

3x y 12 0 Multiply each side by 2. The equation is 3 x y 12 0.

a. Determine an equation of the path of the discus if it is released at (0.96, 0.28).

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2 TRACK AND FIELD Refer to the application at the beginning of the lesson.

Example

b. Will the discus strike an official at (-30, 40)? Explain your answer.

p li c a ti

a. From the figure, we see that p 1, sin 0.28 7 24 or , and cos 0.96 or . 25 25 7 24 An equation of the line is x y 1 0, 25 25

or 24x 7y 25 0.

b. The y-coordinate of the point on the line with 745 an x-coordinate of 30 is or about 106. 7 So the discus will not strike the official at (30, 40).

464

Chapter 7

Trigonometric Identities and Equations

y 1

O 0.96

0.28

We can transform the standard form of a linear equation, Ax By C 0, into normal form if the relationship between the coefficients in the two forms is known. The equations will represent the same line if and only if their A B C corresponding coefficients are proportional. If , then you can cos

sin

solve to find expressions for cos and sin in terms of p and the coefficients. A C cos p B C sin p

Ap Ap or cos C C Bp Bp ➡ sin or sin C C Bp Ap We can divide sin by cos , where cos 0. C C Bp C sin Ap cos C

➡

cos

B A

tan

sin tan cos

Refer to the diagram at the right. Consider an angle in standard position such that

P (A, B)

B A

tan . A 2 B 2

The length of OP A2 B2. is B

So, sin and cos .

2 B2 2 B2 A A B C Since we know that , we can substitute sin p

B C to get the result . p AB2 B2 2 B2 A

Therefore, p . 2 B2 A

If C 0, the sign is chosen so that sin is positive; that is, the same sign as that of B.

The sign is used since p is a measure and must be positive in the equation x cos y sin p 0. Therefore, the sign must be chosen as opposite of the A2 B2, and, if C is negative, use sign of C. That is, if C is positive, use A2 B2. Substitute the values for sin , cos , and p into the normal form. Ax By C 0 2 2 2 2 2 B2 A B A B A

Notice that the standard form is closely related to the normal form.

Changing the Standard Form to Normal Form

The standard form of a linear equation, Ax By C 0, can be changed to normal form by dividing each term by A2 B 2. The sign is chosen opposite the sign of C.

Lesson 7-6

Normal Form of a Linear Equation

465

If the equation of a line is in normal form, you can find the length of the normal, p units, directly from the equation. You can find the angle by using the sin relation tan . However, you must find the quadrant in which the normal cos lies to find the correct angle for . When the equation of a line is in normal form, the coefficient of x is equal to cos , and the coefficient of y is equal to sin . Thus, the correct quadrant can be determined by studying the signs of cos and sin . For example, if sin is negative and cos is positive, the normal lies in the fourth quadrant.

Example

3 Write each equation in normal form. Then find the length of the normal and the angle it makes with the positive x-axis. a. 6x 8y 3 0 Since C is positive, use A2 B2 to determine the normal form. A2 B2 62 82 or 10 6 8 3 3 4 3 10 10 10 5 5 10 4 3 3 Therefore, sin , cos , and p . Since sin and cos are 5 5 10

The normal form is x y 0, or x y 0.

both negative, must lie in the third quadrant. 4 5 4 tan or 3 3 5

233°

sin cos

tan Add 180° to the arctangent to get the angle in Quadrant III.

The normal segment to the origin has length 0.3 unit and makes an angle of 233° with the positive x-axis. b. x 4y 6 0 Since C is negative, use A2 B2 to determine the normal form. A2 B2 (1)2 42 or 17 1

The normal form is x y 0 or

17 17 17 617 417 x 417 17 y 0. Therefore, sin , 17

, and p 617 17 cos . Since sin 0 and cos 0, must lie 17

in the second quadrant. tan

17 4 17 17 17

sin cos

or 4 tan

104°

Add 180° to the arctangent to get the angle in Quadrant II.

617 1.46 units and makes an The normal segment to the origin has length 17 angle of 104° with the positive x-axis.

466

Chapter 7

Trigonometric Identities and Equations

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. Define the geometric meaning of the word normal. 2. Describe how to write the normal form of the equation of a line when p 10

and 30°.

3. Refute or defend the following statement. Determining the normal form of

the equation of a line is like finding the equation of a tangent line to a circle of radius p. 4. Write each form of the equation of a line that you have learned. Compare and

contrast the information that each provides upon inspection. Create a sample problem that would require you to use each form. Guided Practice

Write the standard form of the equation of each line given p, the length of the normal segment, and , the angle the normal segment makes with the positive x-axis. 6. p 3 , 150°

5. p 10, 30°

7 7. p 52 , 4

Write each equation in normal form. Then find the length of the normal and the angle it makes with the positive x-axis. 8. 4x 3y 10

9. y 3x 2

10.

2x 2y 6

11. Transportation

An airport control tower is located at the origin of a coordinate system where the coordinates are measured in miles. An airplane radios in to report its direction and location. The controller determines that the equation of the plane’s path is 3x 4y 8. a. Make a drawing to illustrate the problem. b. What is the closest the plane will come to the tower?

E XERCISES Practice

Write the standard form of the equation of each line given p, the length of the normal segment, and , the angle the normal segment makes with the positive x -axis.

12. p 15, 60° 5 15. p 23 , 6 4 18. p 5, 3

13. p 12, 4 16. p 2, 2 3 19. p , 300° 2

14. p 32 , 135° 17. p 5, 210° 11 20. p 43 , 6

Write each equation in normal form. Then find the length of the normal and the angle it makes with the positive x-axis.

21. 5x 12y 65 0

22. x y 1

23. 3x 4y 15

24. y 2x 4 1 27. y 2 (x 20) 4

25. x 3 x 28. y 4 3

26. 3 x y 2 x y 29. 1 20 24

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Lesson 7-6 Normal Form of a Linear Equation

467

30. Write the standard form of the equation of a line if the point on the line nearest

to the origin is at (6, 8). 31. The point nearest to the origin on a line is at (4, 4). Find the standard form of

the equation of the line.

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Applications and Problem Solving

32. Geometry

The three sides of a triangle are tangent to a unique circle called the incircle. On the coordinate plane, the incircle of ABC has its center at the origin. The lines whose equations are x 4y 617 , 2x 5y 18, and 2 2x y 18 contain the sides of ABC. What is the length of the radius of the incircle?

p li c a ti

33. History

Ancient slingshots were made from straps of leather that cradled a rock until it was released. One would spin the slingshot in a circle, and the initial path of the released rock would be a straight line tangent to the circle at the point of release.

The rock will travel the greatest distance if it is released when the angle between the normal to the path and the horizontal is 45°. The center of the circular path is the origin and the radius of the circle measures 1.25 feet. a. Draw a labeled diagram of the situation. b. Write an equation of the initial path of the rock in standard form.

Consider a line with positive x- and y-intercepts. Suppose makes an angle of with the positive x-axis.

34. Critical Thinking

a. What is the relationship between and , the angle the normal line makes

with the positive x-axis? b. What is the slope of in terms of ? c. What is the slope of the normal line in terms of ? d. What is the slope of in terms of ? 35. Analytic Geometry

Armando was trying to determine how to rotate the graph of a line ° about the origin. He hypothesized that the following would be an equation of the new line. x cos( ) y sin( ) p 0

a. Write the normal form of the line 5x 12y 39 0. b. Determine . Replace by 90° and graph the result. c. Choose another value for , not divisible by 90°, and test Armando’s

conjecture. d. Write an argument as to whether Armando is correct. Include a graph in your

argument. 468

Chapter 7 Trigonometric Identities and Equations

Suppose two lines intersect to form an acute angle . Suppose that each line has a positive y-intercept and that the x-intercepts of the lines are on opposite sides of the origin.

36. Critical Thinking

a. How are the angles 1 and 2 that the respective normals make with the

positive x-axis related?

b. Write an equation involving tan , tan 1, and tan 2. 37. Engineering

The village of Plentywood must build a new water tower to meet the needs of its residents. This means that each of the water mains must be connected to the new tower. On a map of the village, with units in hundreds of feet, the water tower was placed at the origin. Each of the existing water mains can be described by an equation. These equations are 5x y 15, 3x 4y 36, and 5x 2y 20. The cost of laying new underground pipe is $500 per 100 feet. Find the lowest possible cost of connecting the existing water mains to the new tower.

Mixed Review

38. Solve 2 cos2 x 7 cos x 4 0 for 0 x 2. (Lesson 7-5) 1 2 39. If x and y are acute angles such that cos x and cos y , find sin(x y). 6 3

(Lesson 7-3)

40. Graph y sin 4. (Lesson 6-4) 41. Engineering

A metallic ring used in a sprinkler system has a diameter of 13.4 centimeters. Find the length of the metallic cross brace if it subtends a central angle of 26°20 . (Lesson 5-8)

x 17 1 42. Solve . (Lesson 4-6) x5 25 x2 x5

cross brace

43. Manufacturing

A porcelain company produces collectible thimble sets that contain 8 thimbles in a box that is 4 inches by 6 inches by 2 inches. To celebrate the company’s 100th anniversary, they wish to market a deluxe set of 8 large thimbles. They would like to increase each of the dimensions of the box by the same amount to create a new box with a volume that is 1.5 times the volume of the original box. What should be the dimensions of the new box for the large thimbles? (Lesson 4-5)

44. Find the maximum and minimum values of the function f(x, y) 3x y 4

for the polygonal convex set determined by the system of inequalities. (Lesson 2-6) xy8 y3 x2 45. Solve

2 x 0 . (Lesson 2-5) 1 4 3 y 15

46. SAT/ACT Practice A 3

a b

4 5

If , what is the value of 2a b?

B 13

C 14

D 26

E cannot be determined from the given information Extra Practice See p. A39.

Lesson 7-6 Normal Form of a Linear Equation

469

7-7 Distance From a Point to a Line OPTICS

When light waves strike a surface, they are reflected in such a way that the p li c a ti angle of incidence equals the angle of reflection. Consider a flat mirror situated in a coordinate system such that light emanating from the point at (5, 4) strikes the mirror at (3, 0) and then passes through the point at (7, 6). Determine an equation of the line on which the mirror lies. Use the equation to determine the angle the mirror makes with the x-axis. This problem will be solved in Example 4. Ap

• Find the distance from a point to a line. • Find the distance between two parallel lines. • Write equations of lines that bisect angles formed by intersecting lines.

l Wor ea

OBJECTIVES

(5, 4)

(7, 6)

The normal form of a linear equation can be used to find the distance from a point to a line. Let RS be a line in the coordinate plane, and let P(x1, y1) be a point not on RS . P may lie on the same side of RS as the origin does or it may lie on the opposite side. If a line segment joining P to the origin does not intersect RS , point P is on the same side of the line as the origin. Construct TV parallel to RS and passing through P. The distance d between the parallel lines is the distance from P to RS . So that the derivation is valid in both cases, we will use a negative value for d if point P and the origin are on the same side of RS .

T P (x 1, y1) R p

p O

O Graphing Calculator Programs To download a graphing calculator program that computes the distance from a point to a line, visit: www.amc. glencoe.com

d x

x S

V P (x 1, y1)

Let x cos y sin p 0 be the equation of RS in normal form. Since TV is parallel to RS , they have the same slope. The equation for TV can be written as x cos y sin (p d) 0. Solve this equation for d. d x cos y sin p Since P(x1, y1) is on TV , its coordinates satisfy this equation. d x1 cos y1 sin p We can use an equivalent form of this expression to find d when the equation of a line is in standard form.

470

Chapter 7

Trigonometric Identities and Equations

Distance from a Point to a Line

The following formula can be used to find the distance from a point at (x1, y1) to a line with equation Ax By C 0. Ax1 By1 C

d 2 2 A B

The sign of the radical is chosen opposite the sign of C.

The distance will be positive if the point and the origin are on opposite sides of the line. The distance will be treated as negative if the origin is on the same side of the line as the point. If you are solving an application problem, the absolute value of d will probably be required.

Example

1 Find the distance between P(4, 5) and the line with equation 8x 5y 20. First rewrite the equation of the line in standard form. 8x 5y 20 ➡ 8x 5y 20 0 Then use the formula for the distance from a point to a line. Ax1 By1 C

d 2 2 A B

8(4) 5(5) 20

d 2 52 8

A 8, B 5, C 20, x1 4, y1 5

d or about 3.92 Since C is negative, use A2 B2. 89

Therefore, P is approximately 3.92 units from the line 8x 5y 20. Since d is positive, P is on the opposite side of the line from the origin.

You can use the formula for the distance from a point to a line to find the distance between two parallel lines. To do this, choose any point on one of the lines and use the formula to find the distance from that point to the other line.

Example

2 Find the distance between the lines with equations 6x 2y 7 and y 3x 4. Since y 3x 4 is in slope-intercept form, we know that it passes through the point at (0, 4). Use this point to find the distance to the other line. The standard form of the other equation is 6x 2y 7 0. Ax1 By1 C

d 2 2 A B

6(0) 2(4) 7

d 2 ( 6 2)2

A 6, B 2, C 7, x1 0, y1 4

d or about 2.37 Since C is negative, use A2 B2. 40

The distance between the lines is about 2.37 units.

Lesson 7-7

Distance From a Point to a Line

471

An equation of the bisector of an angle formed by two lines in the coordinate plane can also be found using the formula for the distance between a point and a line. The bisector of an angle is the set of all points in the plane equidistant from the sides of the angle. Using this definition, equations of the bisectors of the angles formed by two lines can be found.

y 4

P2 (x 2, y2) d3

d4 d2

d1 P1 (x 1, y1) O

In the figure, 3 and 4 are the bisectors of the angles formed by 1 and 2. P1(x1, y1) is a point on 3, and P2(x2, y2) is a point on 4. Let d1 be the distance from 1 to P1, and let d2 be the distance from 2 to P1. Notice that P1 and the origin lie on opposite sides of 1, so d1 is positive. Since the origin and P1 are on opposite sides of 2, d2 is also positive. Therefore, for any point P1(x1, y1) on 3, d1 d2. However, d3 is positive and d4 is negative. Why? Therefore, for any point P2(x2, y2) on 4, d3 d4. The origin is in the interior of the angle that is bisected by 3, but in the exterior of the angle bisected by 4. So, a good way for you to determine whether to equate distances or to let one distance equal the opposite of the other is to observe the position of the origin.

Relative Position of the Origin

If the origin lies within the angle being bisected or the angle vertical to it, the distances from each line to a point on the bisector have the same sign. If the origin does not lie within the angle being bisected or the angle vertical to it, the distances have opposite signs. To find the equation of a specific angle bisector, first graph the lines. Then, determine whether to equate the distances or to let one distance equal the opposite of the other.

472

Chapter 7

Trigonometric Identities and Equations

Examples

3 Find equations of the lines that bisect the angles formed by the lines 8x 6y 9 and 20x 21y 50. Graph each equation. Note the location of the origin. The origin is in the interior of the acute angle.

[5, 15] scl:1 by [5, 15] scl:1

Bisector of the acute angle

Bisector of the obtuse angle

d1 d2

8x 6y 9

1 1 d1

2 ( 6)2 8

(6)2

20x 21y 50

1 1 d2

20x1 21y1 50 8x1 6y1 9 29 10

202

2 20 (21 )2

(21)2

Simplifying and dropping the subscripts yields 432x 384y 239 and 32x 36y 761, respectively.

4 OPTICS Refer to the application at the beginning of the lesson. l Wor ea

p li c a ti

a. Determine an equation of the line on which the mirror lies. b. Use the equation to determine the angle the mirror makes with the x-axis.

a. Using the points at (5, 4), (7, 6), and (3, 0) and the slope-intercept form, you can find that the light travels along lines with equations 3 9 y 2x 6 and y x . In standard form, 2

y 2x 6

these are 2x y 6 0 and 3x 2y 9 0.

x y 32 x 92

The mirror lies on the angle bisector of the acute angles formed by these lines. The origin is not contained in the angle we wish to bisect, so we use d1 d2.

2x y 6 3x 2y 9 5 13

(2x y 6) 5(3x 2y 9) 13 35x 25 13 y 613 95 0 213 The mirror lies on the line with equation

35 x 25 13 y 613 95 0. 213 Lesson 7-7

Distance From a Point to a Line

473

Multiply by 25 13 2 5 13

to rationalize the denominator.

213 35

b. The slope of the line on which the mirror lies is or 25 13

8 65 . Recall that the slope of a line is the tangent of the angle the line makes with the x-axis. Since tan 8 65 , the positive value for is approximately 93.56°. The mirror makes an angle of 93.56° with the positive x-axis.

C HECK Communicating Mathematics

FOR

U N D E R S TA N D I N G

Read and study the lesson to answer each question. 1. State what is meant by the distance from a point to a line. 2. Tell how to choose the sign of the radical in the denominator of the formula for

the distance from a point to a line. 3. Explain why you can choose any point on either line when calculating the

distance between two parallel lines. 4. Investigate whether the formula for the distance from a point to a line is valid

if the line is vertical or horizontal. Test your conjecture with some specific examples. Explain what is happening when you apply the formula in these situations. Guided Practice

Find the distance between the point with the given coordinates and the line with the given equation. 5. (1, 2), 2x 3y 2

6. (2, 3), 6x y 3

Find the distance between the parallel lines with the given equations. 7. 3x 5y 1

3x 5y 3

1 8. y x 3 3 1 y x 7 3

09. Find equations of the lines that bisect the acute and obtuse angles formed by

the graphs of 6x 8y 5 and 2x 3y 4. 10. Navigation

Juwan drives an ATV due east from the edge of a road into the woods. The ATV breaks down and stops after he has gone 2000 feet. In a coordinate system where the positive y-axis points north and the units are hundreds of feet, the equation of the road is 5x 3y 0. How far is Juwan from the road?

474

Chapter 7 Trigonometric Identities and Equations

www.amc.glencoe.com/self_check_quiz

E XERCISES Find the distance between the point with the given coordinates and the line with the given equation.

Practice

11. (2, 0), 3x 4y 15 0

12. (3, 5), 5x 3y 10 0

13. (0, 0), 2x y 3

2 14. (2, 3), y 4 x 3 4 16. (1, 2), y x 6 3

15. (3, 1), y 2x 5

17. What is the distance from the origin to the graph of 3x y 1 0?

Find the distance between the parallel lines with the given equations. 18. 6x 8y 3

19. 4x 5y 12

6x 8y 5

4x 5y 6

20. y 2x 1

21. y 3x 6

2x y 2

3x y 4

8 22. y x 1 5

3 23. y x 2 3 y 2 x 4

8x 15 5y

24. What is the distance between the lines with equations x y 1 0 and

y x 6?

Find equations of the lines that bisect the acute and obtuse angles formed by the lines with the given equations.

25. 3x 4y 10

5x 12y 26

l Wor ea

Applications and Problem Solving

p li c a ti

26. 4x y 6

15x 8y 68

2 27. y x 1 3

y 3x 2

28. Statistics

Prediction equations are often used to show the relationship between two quantities. A prediction equation relating a person’s systolic blood pressure y to their age x is 4x 3y 228 0. If an actual data point is close to the graph of the prediction equation, the equation gives a good approximation for the coordinates of that point. Classification of Adult a. Linda is 19 years old, and her Systolic Blood Pressure systolic blood pressure is 112. Her father, who is 45 years old, 210 130 has a systolic blood pressure of Very severe Normal 120. For whom is the given hypertension prediction equation a better predictor? 130-139 180-209 b. Refer to the graph at the right. At

what age does the equation begin to predict mild hypertension?

Severe hypertension 160-179 Moderate hypertension

High normal 140-159 Mild hypertension

Source: Archives of Internal Medicine

Lesson 7-7 Distance From a Point to a Line

475

29. Track and Field

In the shot put, the shot must land within a 40° sector. Consider a coordinate system where the vertex of the sector is at the origin, one side lies along the positive x-axis, and the units are meters. If a throw lands at the point with coordinates (16, 12), how far is it from being out-of-bounds?

40˚

30. Critical Thinking

Circle P has its center at (5, 6) and passes through the point at (2, 2). Show that the line with equation 5x 12y 32 0 is tangent to circle P.

31. Geometry

Find the lengths of the altitudes in the triangle with vertices A(3, 4), B(1, 7), and C(1, 3).

Randy Barnes

32. Critical Thinking

In a triangle, the intersection of the angle bisectors is the center of the inscribed circle of the triangle. The inscribed circle is tangent to each side of the triangle. Determine the radius of the inscribed circle (the inradius) for the triangle with vertices at (0, 0), (10, 0), and (4, 12).

Mixed Review

33. Find the normal form of the equation 2x 7y 5. (Lesson 7-6)

3 34. Find cos 2A if sin A . (Lesson 7-4) 6 35. Graph y csc ( 60°). (Lesson 6-7) 36. Aviation

Two airplanes leave an airport at the same time. Each flies at a speed of 110 mph. One flies in the direction 60° east of north. The other flies in the direction 40° east of south. How far apart are the planes after 3 hours? (Lesson 5-8)

37. Physics

The period of a pendulum can be determined by the formula

T 2 , where T represents the period, represents the length of the g pendulum, and g represents acceleration due to gravity. Determine the period of a 2-meter pendulum on Earth if the acceleration due to gravity at Earth’s surface is 9.8 meters per second squared. (Lesson 4-7) 38. Find the value of k in (x2 8x k) (x 2) so that the remainder is zero.

(Lesson 4-3) 39. Solve the system of equations. (Lesson 2-2)

x 2y z 7 2x y z 9 x 3y 2z 10 A

40. SAT/ACT Practice

If the area of square BCED 16 and the area of ABC 6, what is the length of E F ? A 5

B 6

C 7

D 8

E 12

D 476

Chapter 7 Trigonometric Identities and Equations

45˚

Extra Practice See p. A39.

CHAPTER

STUDY GUIDE AND ASSESSMENT VOCABULARY

counterexample (p. 421) difference identity (p. 437) double-angle identity (p. 449) half-angle identity (p. 451) identity (p. 421) normal form (p. 463) normal line (p. 463) opposite-angle identity (p. 426)

principal value (p. 456) Pythagorean identity (p. 423) quotient identity (p. 422) reciprocal identity (p. 422) reduction identity (p. 446) sum identity (p. 437) symmetry identity (p. 424) trigonometric identity (p. 421)

UNDERSTANDING AND USING THE VOCABULARY Choose the letter of the term that best matches each equation or phrase.

1 cos 1. sin 2 2 2. perpendicular to a line, curve, or surface 3. located in Quadrants I and IV for sin x and tan x 4. cos ( ) cos cos sin sin 1 5. cot tan sin 6. tan cos 2 tan 7. tan 2 1 tan2 8. sin (360k° A) sin A 9. 1

cot2

csc2

a. sum identity b. half-angle identity c. normal form d. principal value e. Pythagorean f. g. h. i. j. k.

identity symmetry identity normal line double-angle identity reciprocal identity quotient identity opposite-angle identity

10. uses trigonometry to provide information about a line

For additional review and practice for each lesson, visit: www.amc.glencoe.com Chapter 7 Study Guide and Assessment

477

CHAPTER 7 • STUDY GUIDE AND ASSESSMENT SKILLS AND CONCEPTS OBJECTIVES AND EXAMPLES Lesson 7-1

Identify and use reciprocal identities, quotient identities, Pythagorean identities, symmetry identities, and oppositeangle identities. 1

If is in Quadrant I and cos , find 3 sin . sin2 cos2 1

1 2 sin2 1 3 1 2 sin 1 9 8 2 sin 9 22 sin 3

Use the basic trigonometric identities to verify other identities. Verify that csc x sec x cot x tan x is an identity. csc x sec x cot x tan x 1 1 sin x cos x sin x cos x cos x sin x 1 cos2 x sin2 x sin x cos x sin x cos x 1 1 sin x cos x sin x cos x

Lesson 7-3

Use the sum and difference identities for the sine, cosine, and tangent functions. Find the exact value of sin 105°. sin 105° sin (60° 45°) sin 60° cos 45° cos 60° sin 45° 1 2 3 2 2

6 2

478

Use the given information to determine the trigonometric value. In each case, 0° 90°. 1 11. If sin , find csc . 2 12. If tan 4, find sec . 5 13. If csc , find cos . 3 4 14. If cos , find tan . 5 15. Simplify csc x cos2 x csc x.

Lesson 7-2

REVIEW EXERCISES

Chapter 7 Trigonometric Identities and Equations

Verify that each equation is an identity. 16. cos2 x tan2 x cos2 x 1 1 cos 17. (csc cot )2 1 cos tan sec 1 18. sec 1 tan sin4 x cos4 x 19. 1 cot2 x sin2 x

Use sum or difference identities to find the exact value of each trigonometric function. 20. cos 195° 17 22. sin 12

21. cos 15° 11 23. tan 12

Find each exact value if 0 x

and 0 y . 2

7 2 24. cos (x y) if sin x and cos y 25 3 5 3 25. tan (x y) if tan x and sec y 4 2

CHAPTER 7 • STUDY GUIDE AND ASSESSMENT OBJECTIVES AND EXAMPLES Lesson 7-4

Use the double- and half-angle identities for the sine, cosine, and tangent functions. If is an angle in the first quadrant and 3 sin , find cos 2.

REVIEW EXERCISES Use a half-angle identity to find the exact value of each function. 26. cos 75° 28. sin 22.5°

cos 2 1 2 sin2

1 2 1 8

Lesson 7-5

Solve trigonometric equations and

inequalities. Solve 2 cos2 x 1 0 for 0° x 360°. 2 cos2 x 1 0 1 cos2 x 2

2 cos x 2

x 45°, 135°, 225°, or 315°

7 27. sin 8 29. tan 12

If is an angle in the first quadrant and 3 cos , find the exact value of each 5 function. 30. sin 2

31. cos 2

32. tan 2

33. sin 4

Solve each equation for 0° x 360°. 34. tan x 1 sec x 35. sin2 x cos 2x cos x 0 36. cos 2x sin x 1

Solve each equation for all real values of x.

2 37. sin x tan x tan x 0 2 38. sin 2x sin x 0 39. cos2 x 2 cos x

Lesson 7-6

Write linear equations in normal

form. Write 3x 2y 6 0 in normal form. Since C is negative, use the positive value of A2 B2 . 32 22 13 The normal form is 3 2 6 x y 0 or 13 13 13 3 13 2 13 6 13 x y 0. 13 13 13

Write the standard form of the equation of each line given p, the length of the normal segment, and , the angle the normal segment makes with the positive x-axis. 40. p 23 , 3 2 42. p 3, 3

41. p 5, 90° 43. p 42 , 225°

Write each equation in normal form. Then find the length of the normal and the angle it makes with the positive x-axis. 44. 7x 3y 8 0

45. 6x 4y 5

46. 9x 5y 3

47. x 7y 5

Chapter 7 Study Guide and Assessment

479

CHAPTER 7 • STUDY GUIDE AND ASSESSMENT OBJECTIVES AND EXAMPLES Lesson 7-7

Find the distance from a point to

a line. Find the distance between P(1, 3) and the line with equation 3x 4y 5. Ax By C

1 1 d 2 B2 A

3(1) 4(3) 5

(3)2 42

REVIEW EXERCISES Find the distance between the point with the given coordinates and the line with the given equation. 48. (5, 6), 2x 3y 2 0 49. (3, 4), 2y 3x 6 50. (2, 4), 4y 3x 1 1 51. (21, 20), y x 6 3

20 5

Lesson 7-7

Write equations of lines that bisect angles formed by intersecting lines. Find the equations of the lines that bisect the angles formed by the lines with equations 4x 3y 2 and x 2y 1. 4x1 3y1 2

d1 2 2 (4) 3

4x 3y 2 5

1 1

x 2y 1

1 1 d2

x 2y 1

1 1

Find the distance between the parallel lines with the given equations. x 52. y 6 3 x y 2 3 3 53. y x 3 4 3 1 y x 4 2 54. x y 1

xy5

55. 2x 3y 3 0 2 y x 2 3

The origin is in the interior of one of the obtuse angles formed by the given lines. Bisector of the acute angle: d1 d2 x 2y 1 4x 3y 2 5 5

(45 5)x (35 10)y 25 50

Bisector of the obtuse angle: d1 d2 x 2y 1 4x 3y 2 5 5

(45 5)x (35 10)y 25 50

480

Chapter 7 Trigonometric Identities and Equations

Find the equations of the lines that bisect the acute and obtuse angles formed by the lines with the given equations. 56. y 3x 2 x 3 y 2 2 57. x 3y 2 0 3 y x 3 5

CHAPTER 7 • STUDY GUIDE AND ASSESSMENT APPLICATIONS AND PROBLEM SOLVING 58. Physics

While studying two different physics books, Hector notices that two different formulas are used to find the maximum height of a projectile. One v

60. Surveying

Taigi is surveying a rectangular lot for a new office building. She measures the angle between one side of the lot and the line from her position to the opposite corner of the lot as 30°. She then measures the angle between that line and the line to a telephone pole on the edge of the lot as 45°. If Taigi stands 100 yards from the opposite corner of the lot, how far is she from the telephone pole? (Lesson 7-3)

sin2 2g

0 , and the other is formula is h

v 2 tan2 2g sec

0 h 2 . Are these two formulas

equivalent or is there an error in one of the books? Show your work. (Lesson 7-2)

59. Navigation

Wanda hikes due east from the edge of the road in front of a lodge into the woods. She stops to eat lunch after she has hiked 1600 feet. In a coordinate system where the positive y-axis points north and the units are hundreds of feet, the equation of the road is 4x 2y 0. How far is Wanda from the road? (Lesson 7-7)

100 yd

30˚

Taigi

45˚

Telephone Pole

ALTERNATIVE ASSESSMENT OPEN-ENDED ASSESSMENT

2. Write an equation with sin x tan x on one

side and an expression containing one or more different trigonometric functions on the other side. Verify that your equation is an identity.

PORTFOLIO Choose one of the identities you studied in this chapter. Explain why it is an important identity and how it is used.

Project

trigonometric identities can be used to find exact values for sin , cos , and tan . Find exact values for sin , cos , and tan . Show your work.

Unit 2

1. Give the measure of an angle such that

THE CYBERCLASSROOM

That’s as clear as mud! • Search the Internet to find web sites that have lessons involving trigonometric identities. Find at least two types of identities that were not presented in Chapter 7. • Select one of the types of identities and write at least two of your own examples or sample problems using what you learned on the Internet and in this textbook. • Design your own web page presenting your examples. Use appropriate software to create the web page. Have another student critique your web page.

Additional Assessment practice test.

See p. A62 for Chapter 7

Chapter 7 Study Guide and Assessment

481

SAT & ACT Preparation

CHAPTER

Geometry Problems — Triangles and Quadrilaterals

TEST-TAKING TIP

The ACT and SAT contain problems that deal with triangles, quadrilaterals, lengths, and angles. Be sure to review the properties of isosceles and equilateral triangles. Often several geometry concepts are combined in one problem. Know the number of degrees in various figures. • A straight angle measures 180°. • A right angle measures 90°. • The sum of the measures of the angles in a triangle is 180°. • The sum of the measures of the angles in a quadrilateral is 360°. ACT EXAMPLE

The third side of a triangle cannot be longer than the sum of the other two sides. The third side cannot be shorter than the difference between the other two sides.

SAT EXAMPLE

1. In the figure below, O, N, and M are collinear.

2. If the average measure of two angles in a

If the lengths of ON and NL are the same, the measure of LON is 30°, and LMN is 40°, what is the measure of NLM?

parallelogram is y°, what is the average degree measure of the other two angles?

O 30˚

A 180 y

y B 180 2

D 360 y

E y

HINT 40˚

L A 40° HINT

B 80°

C 90°

M D 120°

E 130°

Look at all the triangles in a figure—large and small.

Solution

Mark the angle you need to find. You may want to label the missing angles and congruent sides, as shown.

30˚

Solution

Look for key words in the problem— average and parallelogram. You need to find the average of two angles. The answer choices are expressions with the variable y. Recall that if the average of two numbers is y, then the sum of the numbers is 2y.

The sum of the angle measures in a parallelogram is 360.

1 2

40˚

Since two sides of ONL are the same length, it is isosceles. So m1 30°. Since the angles in any triangle total 180°, you can write the following equation for OML. 180° 30° 40° (30° m2) 180° 100° m2 180° m2 The answer is choice B. Chapter 7

Underline important words in the problem and the quantity you must find. Look at the answer choices.

So the sum of two angle measures is 2y. Let the sum of the other two angle measures be 2x. Find x.

482

C 360 2y

Trigonometric Identities and Equations

360 2y 2x 360 2(y x) 180 y x x 180 y

Divide each side by 2. Solve for x.

The answer is choice A.

SAT AND ACT PRACTICE After you work each problem, record your answer on the answer sheet provided or on a piece of paper. Multiple Choice 1. In the figure, the measure of A

A 40°

B 60°

80˚

equations y 2x 2 and 7x 3y 11 intersect? B (8, 5) E

2156 , 98

58, 1

x˚

z˚ C 270

D 360

E 450

4. If x y 90° and x and y are positive, then sin x cos y 1 A 1 B 0 C D 1 2 E It cannot be determined from the

information given. 5. In the figure below, what is the sum of the

slopes of AB and A D ? y

E 360 x

(3, 0) O B 0

y A y1 y2 y D (y 1)2

y2 y B (y 1)2 y E y1

y2 y C y1

D 7.5

9. The number of degrees in the sum x° y°

would be the same as the number of degrees in which type of angle?

x˚ A B C D E

y˚

straight angle obtuse angle right angle acute angle It cannot be determined from the information given.

A triangle has a base of 13 units, and the other two sides are congruent. If the side lengths are integers, what is the length of the shortest possible side?

(3, 0) x C 1

E It cannot be determined from the

information given.

y y 7. 2 1 1 2

10. Grid-In

A 1

BCD. What is the length of AE ? C A 5 2 B 6 4 B C 6.5 3 D 7 A E 10

y˚

B 180

b˚

x˚

8. In the figure below, ACE is similar to

3. In the figure below, x y z

A 90

a˚

2. At what point (x, y) do the two lines with

58, 1

D 270 x

D 100° E 120°

A 90 x

C 180 x

C 80°

A (5, 8)

sum a b in terms of x? B 190 x

is 80°. If the measure of B is half the measure of A, what is the measure of C?

6. In the rectangle PQRS below, what is the

D 6 SAT/ACT Practice For additional test practice questions, visit: www.amc.glencoe.com SAT & ACT Preparation

483

[PDF] TRIGONOMETRIC IDENTITIES AND EQUATIONS - Free Download PDF (2024)

References