Section 6-2 Sum, Difference, and Cofunction Identities

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6-2 Sum, Difference, and Cofunction Identities

Section 6-2 Sum, Difference, and Cofunction Identities Sum and Difference Identities for Cosine Cofunction Identities Sum and Difference Identities for Sine and Tangent Summary and Use

The basic identities discussed in Section 6-1 involved only one variable. In this section, we consider identities that involve two variables.

Sum and Difference Identities for Cosine We start with the important difference identity for cosine: cos (x ⴚ y) ⴝ cos x cos y ⴙ sin x sin y

(1)

Many other useful identities can be readily verified from this particular one. Here, we sketch a proof of equation (1) assuming x and y are in the interval (0, 2␲) and x ⬎ y ⬎ 0. Identity (1) holds, however, for all real numbers and angles in radian or degree measure. First, associate x and y with arcs and angles on the unit circle as indicated in Figure 1(a). Using the definitions of the circular functions given in Section 5-2, label the terminal points of x and y as shown in Figure 1(a). FIGURE 1

e f C (cos (x ⫺ y), sin (x ⫺ y))

Difference identity. a b A(cos y, sin y)

x y 1

1 O

x⫺y

x⫺y

D(1, 0)

O

1

D(1, 0)

1 c d B (cos x, sin x) (a)

(b)

Now if you rotate the triangle AOB clockwise about the origin until the terminal point A coincides with D(1, 0), then terminal point B will be at C, as shown in Figure 1(b). Thus, since rotation preserves lengths, d(A, B) ⫽ d(C, D) 兹(c ⫺ a)2 ⫹ (d ⫺ b)2 ⫽ 兹(1 ⫺ e)2 ⫹ (0 ⫺ f)2 (c ⫺ a)2 ⫹ (d ⫺ b)2 ⫽ (1 ⫺ e)2 ⫹ f 2 c2 ⫺ 2ac ⫹ a2 ⫹ d 2 ⫺ 2db ⫹ b2 ⫽ 1 ⫺ 2e ⫹ e2 + f 2 (c2 ⫹ d 2) ⫹ (a2 ⫹ b2) ⫺ 2ac ⫺ 2db ⫽ 1 ⫺ 2e ⫹ (e2 ⫹ f 2)

(2)

Since points A, B, and C are on unit circles, c2 ⫹ d 2 ⫽ 1, a2 ⫹ b2 ⫽ 1, and e2 ⫹ f 2 ⫽ 1, and equation (2) simplifies to e ⫽ ac ⫹ bd

(3)

462

6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS

Replacing e, a, c, b, and d with cos (x ⫺ y), cos y, cos x, sin y, and sin x, respectively (see Fig. 1), we obtain cos (x ⫺ y) ⫽ cos y cos x ⫹ sin y sin x ⫽ cos x cos y ⫹ sin x sin y

(4)

We have thus established the difference identity for cosine. If we replace y with ⫺y in equation (4) and use the identities for negatives (a good exercise for you), we obtain cos (x ⴙ y) ⴝ cos x cos y ⴚ sin x sin y

(5)

This is the sum identity for cosine.

Explore/Discuss

1

Discuss how you would show that, in general, cos (x ⫺ y) ⫽ cos x ⫺ cos y and cos (x ⫹ y) ⫽ cos x ⫹ cos y

Cofunction Identities To obtain sum and difference identities for the sine and tangent functions, we first derive cofunction identities directly from equation (1), the difference identity for cosine: cos (x ⫺ y) ⫽ cos x cos y ⫹ sin x sin y cos

冢␲2 ⫺ y冣 ⫽ cos ␲2 cos y ⫹ sin ␲2 sin y ⫽ (0)(cos y) ⫹ (1)(sin y) ⫽ sin y

Thus, we have the cofunction identity for cosine: cos

冢␲2 ⴚ y冣 ⴝ sin y

(6)

for y any real number or angle in radian measure. If y is in degree measure, replace ␲/2 with 90°. Now, if we let y ⫽ ␲/2 ⫺ x in equation (6), we have cos

冤 ␲2 ⫺ 冢␲2 ⫺ x冣冥 ⫽ sin 冢␲2 ⫺ x冣 cos x ⫽ sin

冢␲2 ⫺ x冣


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This is the cofunction identity for sine; that is, sin

冢 2 ⴚ x冣 ⴝ cos x ␲

(7)

where x is any real number or angle in radian measure. If x is in degree measure, replace ␲/2 with 90°. Finally, we state the cofunction identity for tangent (and leave its derivation to Problem 10 in Exercise 6-2): tan

冢␲2 ⴚ x冣 ⴝ cot x

(8)

for x any real number or angle in radian measure. If x is in degree measure, replace ␲/2 with 90°. Remark

If 0 ⬍ x ⬍ 90°, then x and 90° ⫺ x are complementary angles. Originally, “cosine,” “cotangent,” and “cosecant” meant, respectively, “complements sine,” “complements tangent,” and “complements secant.” Now we simply refer to cosine, cotangent, and cosecant as cofunctions of sine, tangent, and secant, respectively.

Sum and Difference Identities for Sine and Tangent To derive a difference identity for sine, we use equations (1), (6), and (7) as follows: sin (x ⫺ y) ⫽ cos

冤 ␲2 ⫺ (x ⫺ y)冥

⫽ cos

冤冢␲2 ⫺ x冣 ⫺ (⫺y)冥

⫽ cos

冢␲2 ⫺ x冣 cos (⫺y) ⫹ sin 冢␲2 ⫺ x冣 sin (⫺y)

⫽ sin x cos y ⫺ cos x sin y

Use equation (6). Algebra Use equation (1). Use equations (6) and (7) and identities for negatives.

The same result is obtained by replacing ␲/2 with 90°. Thus, sin (x ⴚ y) ⴝ sin x cos y ⴚ cos x sin y

(9)

is the difference identity for sine. Now, if we replace y in equation (9) with ⫺y (a good exercise for you), we obtain sin (x ⴙ y) ⴝ sin x cos y ⴙ cos x sin y the sum identity for sine.

(10)

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It is not difficult to derive sum and difference identities for the tangent function. See if you can supply the reason for each step: tan (x ⫺ y) ⫽ ⫽

sin (x ⫺ y) cos (x ⫺ y) sin x cos y ⫺ cos x sin y cos x cos y ⫹ sin x sin y

cos x sin y sin x cos y ⫺ cos x cos y cos x cos y ⫽ cos x cos y sin x sin y ⫹ cos x cos y cos x cos y

Divide the numerator and denominator by cos x and cos y.

sin x sin y ⫺ cos x cos y ⫽ sin x sin y 1⫹ cos x cos y ⫽

tan x ⫺ tan y 1 ⫹ tan x tan y

Thus, tan (x ⴚ y) ⴝ

tan x ⴚ tan y 1 ⴚ tan x tan y

(11)

for all angles or real numbers x and y for which both sides are defined. This is the difference identity for tangent. If we replace y in equation (11) with ⫺y (another good exercise for you), we obtain tan (x ⴙ y) ⴝ

tan x ⴙ tan y 1 ⴚ tan x tan y

(12)

the sum identity for tangent.

Explore/Discuss

2

Discuss how you would show that, in general, tan (x ⫺ y) ⫽ tan x ⫺ tan y and tan (x ⫹ y) ⫽ tan x ⫹ tan y

Summary and Use Before proceeding with examples illustrating the use of these new identities, review the list given in the following box.


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SUMMARY OF IDENTITIES Sum Identities sin (x ⫹ y) ⫽ sin x cos y ⫹ cos x sin y cos (x ⫹ y) ⫽ cos x cos y ⫺ sin x sin y tan (x ⫹ y) ⫽

tan x ⫹ tan y 1 ⫺ tan x tan y

Difference Identities sin (x ⫺ y) ⫽ sin x cos y ⫺ cos x sin y cos (x ⫺ y) ⫽ cos x cos y ⫹ sin x sin y tan (x ⫺ y) ⫽

tan x ⫺ tan y 1 ⫹ tan x tan y

Cofunction Identities (Replace ␲/2 with 90° if x is in degrees.) cos

EXAMPLE

1 Solution

冢␲2 ⫺ x冣 ⫽ sin x

sin

冢␲2 ⫺ x冣 ⫽ cos x

tan

冢␲2 ⫺ x冣 ⫽ cot x

Using the Difference Identity Simplify cos (x ⫺ ␲) using the difference identity. cos (x ⫺ y) ⫽ cos x cos y ⫹ sin x sin y cos (x ⫺ ␲) ⫽ cos x cos ␲ ⫹ sin x sin ␲ ⫽ (cos x) (⫺1) ⫹ (sin x) (0) ⫽ ⫺cos x

MATCHED PROBLEM

Simplify sin (x ⫹ 3␲/2) using a sum identity.

1 EXAMPLE

2 Solution

Checking the Use of Sum and Difference Identities on a Graphing Utility Simplify sin (x ⫺ ␲) using a difference identity. Enter the original form as y1 and the converted form as y2 in a graphing utility, then graph both in the same viewing window. sin (x ⫺ y) ⫽ sin x cos y ⫺ cos x sin y sin (x ⫺ ␲) ⫽ sin x cos ␲ ⫺ cos x sin ␲ ⫽ (sin x) (⫺1) ⫺ (cos x) (0) ⫽ ⫺sin x

466


Graph y1 ⫽ sin (x ⫺ ␲) and y2 ⫽ ⫺sin x in the same viewing window (Fig. 2). Use TRACE and move back and forth between y1 and y2 for different values of x to see that the corresponding y values are the same, or nearly the same. FIGURE 2

4

⫺2␲

2␲

⫺4

MATCHED PROBLEM

2 EXAMPLE

3 Solution

Simplify cos (x ⫹ 3␲/2) using a sum identity. Enter the original form as y1 and the converted form as y2 in a graphing utility, then graph both in the same viewing window.

Finding Exact Values Find the exact value of tan 75° in radical form. Since we can write 75° ⫽ 45° ⫹ 30°, the sum of two special angles, we can use the sum identity for tangents with x ⫽ 45° and y ⫽ 30°: tan (x ⫹ y) ⫽

tan x ⫹ tan y 1 ⫺ tan x tan y tan 45° ⫹ tan 30° 1 ⫺ tan 45° tan 30°

Sum identity

⫽

1 ⫹ (1/兹3) 1 ⫺ 1(1/兹3)

Evaluate functions exactly.

⫽

兹3 ⫹ 1 兹3 ⫺ 1

tan (45° ⫹ 30°) ⫽

⫽ 2 ⫹ 兹3 MATCHED PROBLEM

Multiply numerator and denominator by 兹3 and simplify. Rationalize denominator and simplify.

Find the exact value of cos 15° in radical form.

3 EXAMPLE

4 Solution

Finding Exact Values Find the exact value of cos (x ⫹ y), given sin x ⫽ 35, cos y ⫽ 45, x is an angle in quadrant II, and y is an angle in quadrant I. Do not use a calculator. We start with the sum identity for cosine, cos (x ⫹ y) ⫽ cos x cos y ⫺ sin x sin y We know sin x and cos y but not cos x and sin y. We find the latter two using two different methods as follows (use the method that is easiest for you).

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Given sin x ⫽

3 5

and x is an angle in quadrant II, find cos x:

Method I. Use a reference triangle:

Method II. Use a unit circle:

b

b P 冢a,

(a, 3) 5

3 5

冣

x

x

3

a (1, 0)

a

a

a 5

cos x ⫽ a

a2 ⫹ 32 ⫽ 52

a2 ⫹ ( 35)2 ⫽ 1

a2 ⫽ 16

a2 ⫽ 16 25

cos x ⫽

a ⫽ ⫾45

a ⫽ ⫾4 a ⫽ ⫺4

In quadrant II,

cos x ⫽ ⫺ 45

Therefore, Given cos y ⫽

4 5

cos x ⫽ ⫺ 45

Therefore,

and y is an angle in quadrant I, find sin y:

Method I. Use a reference triangle: b

a ⫽ ⫺ 45

In quadrant II,

Method II. Use a unit circle: b

(4, b)

P 冢 5 , b冣 4

5 y

b

x

a

a (1, 0)

4

b 5

sin y ⫽ b

42 ⫹ b2 ⫽ 52

( 45)2 ⫹ b2 ⫽ 1

sin y ⫽

9 b2 ⫽ 25

b2 ⫽ 9

b ⫽ ⫾35

b ⫽ ⫾3 In quadrant I, Therefore,

b⫽3 sin y ⫽ 35

In quadrant I, Therefore,

b ⫽ 35 sin y ⫽ 35

We can now evaluate cos (x ⫹ y) without knowing x and y: cos (x ⫹ y) ⫽ cos x cos y ⫺ sin x sin y ⫽ (⫺ 45)( 45) ⫺ ( 35)( 35) ⫽ ⫺ 25 25 ⫽ ⫺1 MATCHED PROBLEM

4 EXAMPLE

5

Find the exact value of sin (x ⫺ y), given sin x ⫽ ⫺ 23, cos y ⫽ 兹5/3, x is an angle in quadrant III, and y is an angle in quadrant IV. Do not use a calculator.

Identity Verification cos (x ⫺ y) Verify the identity tan x ⫹ cot y ⫽ . cos x sin y

468


cos (x ⫺ y) cos x cos y ⫹ sin x sin y ⫽ cos x sin y cos x sin y

Verification

⫽

cos x cos y sin x sin y ⫹ cos x sin y cos x sin y

⫽ cot y ⫹ tan x

Difference identity for cosine Algebra Quotient identities

⫽ tan x ⫹ cot y

Verify the identity cot y ⫺ cot x ⫽

MATCHED PROBLEM

5

sin (x ⫺ y) . sin x sin y

Answers to Matched Problems 1. ⫺cos x

2. y1 ⫽ cos (x ⫹ 3␲/2), y2 ⫽ sin x

3. (1 ⫹ 兹3)/2兹2 or (兹6 ⫹ 兹2)/4

4. ⫺4兹5/9

4

⫺2␲

2␲

⫺4

5.

sin (x ⫺ y) sin x cos y ⫺ cos x sin y sin x cos y cos x sin y ⫽ ⫽ ⫺ ⫽ cot y ⫺ cot x sin x sin y sin x sin y sin x sin y sin x sin y

EXERCISE 6-2

11. csc

A We can use sum identities to establish periodic properties for the trigonometric functions. Verify the identities in Problems 1–8 using the sum identities. 1. sin (x ⫹ 2␲) ⫽ sin x

2. cos (x ⫹ 2␲) ⫽ cos x

3. tan (x ⫹ ␲) ⫽ tan x

4. cot (x ⫹ ␲) ⫽ cot x

冢 2 ⫺ x冣 ⫽ sec x ␲

12. sec

冢 2 ⫺ x冣 ⫽ csc x ␲

Convert Problems 13–18 to forms involving sin x, cos x, and/or tan x using sum or difference identities. 13. sin (30° ⫺ x)

14. sin (x ⫺ 45°)

15. sin (180° ⫺ x)

16. cos (x ⫹ 180°)

冢

17. tan x ⫹

␲ 3

冣

18. tan

冢␲4 ⫺ x冣

5. cos (x ⫹ 2k␲) ⫽ cos x, k an integer 6. sin (x ⫹ 2k␲) ⫽ sin x, k an integer 7. cot (x ⫹ k␲) ⫽ cot x, k an integer

Use appropriate identities to find exact values for Problems 19–26. Do not use a calculator.

8. tan (x ⫹ k␲) ⫽ tan x, k an integer Verify each identity in Problems 9–12 using cofunction identities for sine and cosine and basic identities discussed in Section 6-1. 9. cot

冢␲2 ⫺ x冣 ⫽ tan x

10. tan

B

冢␲2 ⫺ x冣 ⫽ cot x

19. sec 75°

20. sin 75°

21. sin

7␲ 12

␲ ␲ ⫽ ⫹ 冥冤Hint: 7␲ 12 3 4

22. cos

␲ 12

冤Hint: 12␲ ⫽ ␲4 ⫺ ␲6 冥

6-2 Sum, Difference, and Cofunction Identities 23. cos 74° cos 44° ⫹ sin 74° sin 44° 24. sin 22° cos 38° ⫹ cos 22° sin 38° 25.

tan 27° ⫹ tan 18° 1 ⫺ tan 27° tan 18°

26.

tan 110° ⫺ tan 50° 1 ⫹ tan 110° tan 50°

Find sin (x ⫺ y) and tan (x ⫹ y) exactly without a calculator using the information given in Problems 27–30. 27. sin x ⫽ ⫺ 35, sin y ⫽ 兹8/3, x is a quadrant IV angle, y is a quadrant I angle. 28. sin x ⫽ 23, cos y ⫽ ⫺ 14, x is a quadrant II angle, y is a quadrant III angle. 29. tan x ⫽ 34, tan y ⫽ ⫺ 12, x is a quadrant III angle, y is a quadrant IV angle. 30. cos x ⫽ ⫺ 13, tan y ⫽ 12, x is a quadrant II angle, y is a quadrant III angle.

Verify each identity in Problems 31–44. 31. cos 2x ⫽ cos2 x ⫺ sin2 x 33. cot (x ⫹ y) ⫽

32. sin 2x ⫽ 2 sin x cos x

46. x ⫽ 3.042, y ⫽ 2.384

47. x ⫽ 42.08°, y ⫽ 68.37°

48. x ⫽ 128.3°, y ⫽ 25.62°

49. Explain how you would show that, in general, sec (x ⫺ y) ⫽ sec x ⫺ sec y 50. Explain how you would show that, in general, csc (x ⫹ y) ⫽ csc x ⫹ csc y In Problems 51–56, use sum or difference identities to convert each equation to a form involving sin x, cos x, and/or tan x. Enter the original equation in a graphing utility as y1 and the converted form as y2, then graph y 1 and y 2 in the same viewing window. Use TRACE to compare the two graphs. 51. y ⫽ sin (x ⫹ ␲/6)

52. y ⫽ sin (x ⫺ ␲/3)

53. y ⫽ cos (x ⫺ 3␲/4)

54. y ⫽ cos (x ⫹ 5␲/6)

55. y ⫽ tan (x ⫹ 2␲/3)

56. y ⫽ tan (x ⫺ ␲/4)

57. sin [cos⫺1 (⫺ 45) ⫹ sin⫺1 (⫺ 35)] 58. cos [sin⫺1 (⫺ 35) ⫹ cos⫺1 ( 45)] 59. sin [arccos 12 ⫹ arcsin (⫺1)]

cot2 x ⫺ 1 36. cot 2x ⫽ 2 cot x

60. cos [arccos (⫺兹3/2) ⫺ arcsin (⫺ 12)] 61. Express sin (sin⫺1 x ⫹ cos⫺1 y) in an equivalent form free of trigonometric and inverse trigonometric functions.

sin (v ⫹ u) cot u ⫹ cot v ⫽ 37. sin (v ⫺ u) cot u ⫺ cot v

62. Express cos (sin⫺1 x ⫺ cos⫺1 y) in an equivalent form free of trigonometric and inverse trigonometric functions.

sin (u ⫹ v) tan u ⫹ tan v ⫽ 38. sin (u ⫺ v) tan u ⫺ tan v cos (x ⫹ y) sin x cos y

Verify the identities in Problems 63 and 64. 63. cos (x ⫹ y ⫹ z) ⫽ cos x cos y cos z ⫺ sin x sin y cos z ⫺ sin x cos y sin z ⫺ cos x sin y sin z

sin (x ⫺ y) 40. tan x ⫺ tan y ⫽ cos x cos y

64. sin (x ⫹ y ⫹ z) ⫽ sin x cos y cos z ⫹ cos x sin y cos z ⫹ cos x cos y sin z ⫺ sin x sin y sin z

cot y ⫺ cot x 41. tan (x ⫺ y) ⫽ cot x cot y ⫹ 1 cot x ⫹ cot y 42. tan (x ⫹ y) ⫽ cot x cot y ⫺ 1

冢冣 cos h ⫺ 1 sin (x ⫹ h) ⫺ sin x ⫽ sin x 冢 44. 冣 ⫹ cos x 冢sinh h 冣 h h 43.

45. x ⫽ 5.288, y ⫽ 1.769

In Problems 57–60, evaluate exactly as real numbers without the use of a calculator.

2 tan x 1 ⫺ tan2 x

39. cot x ⫺ tan y ⫽

Evaluate both sides of the difference identity for sine and the sum identity for tangent for the values of x and y indicated in Problems 45–48. Evaluate to four significant digits using a calculator.

C

cot x cot y ⫺ 1 cot x ⫹ cot y

cot x cot y ⫹ 1 34. cot (x ⫺ y) ⫽ cot y ⫺ cot x 35. tan 2x ⫽

469

冢

冣

cos (x ⫹ h) ⫺ cos x cos h ⫺ 1 sin h ⫽ cos x ⫺ sin x h h h

In Problems 65 and 66, write each equation in terms of a single trigonometric function. Enter the original equation in a graphing utility as y1 and the converted form as y 2 , then graph y1 and y2 in the same viewing window. Use TRACE to compare the two graphs. 65. y ⫽ cos 1.2x cos 0.8x ⫺ sin 1.2x sin 0.8x 66. y ⫽ sin 0.8x cos 0.3x ⫺ cos 0.8x sin 0.3x

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6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 70. Light Refraction. Use the results of Problem 69 to find ␤ to the nearest degree if ␣ ⫽ 43°, M ⫽ 0.25 inch, and N ⫽ 0.11 inch.

APPLICATIONS 67. Analytic Geometry. Use the information in the figure to show that m2 ⫺ m1 tan (␪2 ⫺ ␪1) ⫽ 1 ⫹ m1m2 L2

␪2 ⫺ ␪1 L1

␪2

␪1

tan ␪1 ⫽ Slope of L1 ⫽ m1

★

71. Surveying. El Capitan is a large monolithic granite peak that rises straight up from the floor of Yosemite Valley in Yosemite National Park. It attracts rock climbers worldwide. At certain times, the reflection of the peak can be seen in the Merced River that runs along the valley floor. How can the height H of El Capitan above the river be determined by using only a sextant h feet high to measure the angle of elevation, ␤, to the top of the peak, and the angle of depression, ␣, of the reflected peak top in the river? (See accompanying figure, which is not to scale.) (A) Using right triangle relationships, show that ␤ cot ␣ 冢11 ⫹⫺ tan tan ␤ cot ␣ 冣

tan ␪2 ⫽ Slope of L2 ⫽ m2

H⫽h

68. Analytic Geometry. Find the acute angle of intersection between the two lines y ⫽ 3x ⫹ 1 and y ⫽ 12x ⫺ 1. (Use the results of Problem 67.) ★★

69. Light Refraction. Light rays passing through a plate glass window are refracted when they enter the glass and again when they leave to continue on a path parallel to the entering rays (see the figure). If the plate glass is M inches thick, the parallel displacement of the light rays is N inches, the angle of incidence is ␣, and the angle of refraction is ␤, show that tan ␤ ⫽ tan ␣ ⫺

(B) Using sum or difference identities, show that the result in part A can be written in the form (␣ ⫹ ␤) 冤 sin sin (␣ ⫺ ␤)冥

H⫽h

(C) If a sextant of height 4.90 feet measures ␣ to be 46.23° and ␤ to be 46.15°, compute the height H of El Capitan above the Merced River to three significant digits.

N sec ␣ M

El Capitan

[Hint: First use geometric relationships to obtain

D

N M ⫽ sec (90° ⫺ ␤) sin (␣ ⫺ ␤) then use difference identities and fundamental identities to complete the derivation.] M H Air ␣ ␤

E⬘ ␤ E

Air

h Plate glass

A N

␣ ␣

␣ B Merced River

C Yosemite National Park