Download SECTION 6-2 Sum, Difference, and Cofunction Identities

6-2
0 x /2. Simplify, using a basic identity, and write
the final form free of radicals.
98. In the radical form a 2 u 2, a 0, let u a cos x,
0 x . Simplify, using a basic identity, and write the final form free of radicals.
100. In the radical form a 2 u 2, a 0, let u a cot x, 0 x /2. Simplify, using a basic identity, and write the final form free of radicals.
99. In the radical form a 2 u 2, a 0, let u a tan x,
SECTION
6-2
461
Sum, Difference, and Cofunction Identities
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 y) cos x cos y 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. It then follows easily, by periodicity and basic identities, that
(1) holds 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 Difference identity.
a
b
A(cos y, sin y)
x
y
xy
1
1
O
xy
e
f
C (cos (x y), sin (x y))
O
D(1, 0)
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)
(c a)2 (d b)2 (1 e)2 (0 f )2
(c a)2 (d b)2 (1 e)2 f 2
462
6 Trigonometric Identities and Conditional Equations
c 2 2ac a 2 d 2 2db b 2 1 2e e 2 f 2
(c 2 d 2) (a 2 b 2) 2ac 2db 1 2e (e 2 f 2)
(2)
Since points A, B, and C are on unit circles, c 2 d 2 1, a 2 b 2 1, and e 2 f 2 1, and equation (2) simplifies to
e ac bd
(3)
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 y) cos x cos y 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
2 y 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
2 y sin y
(6)
for y any real number or angle in radian measure. If y is in degree measure, replace
/2 with 90°.
6-2
Sum, Difference, and Cofunction Identities
463
Now, if we let y /2 x in equation (6), we have
cos
2 2 x sin 2 x
cos x sin
2 x
This is the cofunction identity for sine; that is,
sin
2 x 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 12 in Exercise 6-2):
tan
2 x 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:
2 (x y)
Use equation (6).
cos
2 x (y)
Algebra
cos
2 x cos (y) sin 2 x sin (y)
Use equation (1).
sin (x y) cos
sin x cos y cos x sin y
Use equations (6)
and (7) and identities
for negatives.
The same result is obtained by replacing /2 with 90°. Thus,
sin (x y) sin x cos y 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 y) sin x cos y cos x sin y
the sum identity for sine.
(10)
464
6 Trigonometric Identities and Conditional Equations
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
sin x cos y
cos x sin 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 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, for all angles or real numbers x and y,
tan (x y) tan x tan y
1 tan x tan y
(11)
is the difference identity for tangent.
If we replace y in equation (11) with y (another good exercise for you), we
obtain
tan (x y) tan x tan y
1 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.
6-2
Sum, Difference, and Cofunction Identities
465
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
2 x sin x
sin
2 x cos x
tan
2 x cot x
Using the Difference Identity
Simplify cos (x ) using the difference identity.
Solution
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 1
EXAMPLE 2
Simplify sin (x 3/2) using a sum identity.
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.
466
6 Trigonometric Identities and Conditional Equations
Solution
4
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)
2
sin x
2
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.
4
FIGURE 2
Matched Problem 2
EXAMPLE 3
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 value of tan 75° in exact radical form.
Solution
Since we can write 75° 45° 30°, the sum of two special angles, we can use the
sum identity for tangent with x 45° and y 30°:
tan (x y) tan 45° tan 30°
1 tan 45° tan 30°
Sum identity
1 (1/3)
1 1(1/3)
Evaluate functions exactly.
3 1
3 1
Multiply numerator and denominator
by 3 and simplify.
2 3
Rationalize denominator and simplify.
tan (45° 30°) Matched Problem 3
EXAMPLE 4
tan x tan y
1 tan x tan y
Find the value of cos 15° in exact radical form.
Finding Exact Values
Find the exact value of cos (x y), given sin x 53, cos y 45, x is an angle in
quadrant II, and y is an angle in quadrant I. Do not use a calculator or table.
Solution
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).
6-2
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 a,
(a, 3)
5
3
5
x
x
3
a
(1, 0)
a
a
a
5
cos x a
a 2 32 52
a 2 ( 35 )2 1
a 2 16
a 2 16
25
cos x a 4
a 4
In quadrant II,
cos x 45
Therefore,
Given cos y 4
5
b
a 45
a 45
In quadrant II,
cos x 45
Therefore,
and y is an angle in quadrant I, find sin y:
Method I. Use a reference triangle:
Method II. Use a unit circle:
b
(4, b)
P 5 , b
4
5
y
b
x
a
(1, 0)
a
4
sin y b
5
sin y b
42 b 2 5 2
( 45 )2 b 2 1
b2 9
9
b 2 25
b 3
In quadrant I,
467
Sum, Difference, and Cofunction Identities
b3
b 35
In quadrant I,
Therefore,
Therefore,
sin y 35
We can now evaluate cos (x y) without knowing x and y:
b 35
sin y 35
cos (x y) cos x cos y sin x sin y
( 45 )( 45 ) ( 35 )( 35 ) 25
25 1
Matched Problem 4
Find the exact value of sin (x y), given sin x 23, cos y 5/3, x is an angle
in quadrant III, and y is an angle in quadrant IV. Do not use a calculator or table.
468
6 Trigonometric Identities and Conditional Equations
EXAMPLE 5
Identity Verification
Verify the identity: tan x cot y cos (x y)
cos x sin y
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
Matched Problem 5
Verify the identity: cot y cot x sin (x y)
sin x sin y
Answers to Matched Problems
1. cos x
2. y1 cos (x 3/2), y2 sin x
4
2
2
4
3. (1 3)/22 or (6 2)/4
4. 45/9
sin (x y) sin x cos y cos x sin y sin x cos y cos x sin y
5.
cot y cot x
sin x sin y
sin x sin y
sin x sin y
sin x sin y
EXERCISE
6-2
A
In Problems 1–10, is the equation an identity? Explain,
making use of sum or difference identities.
1. sin (x 2) sin x
2. cos (x 2) cos x
3. cos (x ) cos x
4. sin (x ) sin x
5. tan (x ) tan x
6. tan ( x) tan x
7. csc (2 x) csc x
8. sec (2 x) sec x
9. sin (x 2k) sin x, k an integer
10. tan (x k) tan x, k an integer
Verify each identity in Problems 11–14 using cofunction
identities for sine and cosine and basic identities discussed
in Section 6-1.
2 x tan x
13. csc x sec x
2
11. cot
2 x cot x
14. sec x csc x
2
12. tan
Convert Problems 15–20 to forms involving sin x, cos x,
and/or tan x using sum or difference identities.
15. cos (x 45°)
16. sin (x 30°)
6-2
3 x
Sum, Difference, and Cofunction Identities
sin (x y)
cos x cos y
18. cos ( x)
42. tan x tan y 20. tan (x 45°)
43. tan (x y) cot y cot x
cot x cot y 1
B
44. tan (x y) cot x cot y
cot x cot y 1
Use appropriate identities to find exact values for Problems
21–28. Do not use a calculator.
45.
17. tan
19. sin (x 90°)
21. cos 20° cos 25° sin 20° sin 25°
22. sin 75° cos 15° cos 75° sin 15°
23.
tan 50° tan 20°
1 tan 50° tan 20°
24.
25. sin 15°
27. cos
tan 35° tan 25°
1 tan 35° tan 25°
26. cos 15°
11
11 2 Hint:
12
12
3
4
12 Hint: 12 6 4 28. sin 469
cos h 1
sin (x h) sin x
46.
sin x cos x sinh h h
h
cos (x h) cos x
cos h 1
sin h
sin x
cos x
h
h
h
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 47–50. Evaluate to 4 significant digits using a
calculator.
47. x 5.288, y 1.769
48. x 3.042, y 2.384
49. x 42.08°, y 68.37°
50. x 128.3°, y 25.62°
51. Explain how you would show that, in general,
Find sin (x y) and tan (x y) exactly without a calculator using the information given in Problems 29–32.
sec (x y) sec x sec y
52. Explain how you would show that, in general,
29. sin x sin y 8/3, x is a quadrant IV angle, y is a
quadrant I angle.
35,
30. sin x 23, cos y 14, x is a quadrant II angle, y is a quadrant III angle.
31. tan x 43, tan y 12, x is a quadrant III angle, y is a quadrant IV angle.
32. cos x 13, tan y 12, x is a quadrant II angle, y is a quadrant III angle.
csc (x y) csc x csc y
In Problems 53–58, 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 y1 and y2 in
the same viewing window. Use TRACE to compare the two
graphs.
53. y sin (x /6)
54. y sin (x /3)
Verify each identity in Problems 33–46.
55. y cos (x 3/4)
56. y cos (x 5/6)
33. cos 2x cos2 x sin2 x
57. y tan (x 2/3)
58. y tan (x /4)
34. sin 2x 2 sin x cos x
35. cot (x y) cot x cot y 1
cot x cot y
C
36. cot (x y) cot x cot y 1
cot y cot x
In Problems 59–62, evaluate exactly as real numbers without the use of a calculator.
37. tan 2x 2 tan x
1 tan2 x
38. cot 2x cot2 x 1
2 cot x
59. sin [cos1 ( 45) sin1 ( 35)]
60. cos [sin1 ( 35 ) cos1 ( 45 )]
39.
sin (v u) cot u cot v
sin (v u) cot u cot v
61. sin [arccos 12 arcsin (1)]
40.
sin (u v) tan u tan v
sin (u v) tan u tan v
63. Express sin (sin1 x cos1 y) in an equivalent form free
of trigonometric and inverse trigonometric functions.
cos (x y)
sin x cos y
64. Express cos (sin1 x cos1 y) in an equivalent form free
of trigonometric and inverse trigonometric functions.
41. cot x tan y 62. cos [arccos (3/ 2) arcsin ( 12 )]
470
6 Trigonometric Identities and Conditional Equations
Verify the identities in Problems 65 and 66.
tan tan 65. 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
N
sec M
[Hint: First use geometric relationships to obtain
66. 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
N
M
sin (90° ) sin ( )
In Problems 67–68, 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 y2, then
graph y1 and y2 in the same viewing window. Use TRACE
to compare the two graphs.
then use difference identities and fundamental identities to
complete the derivation.]
67. y cos 1.2x cos 0.8x sin 1.2x sin 0.8x
★
68. y sin 0.8x cos 0.3x cos 0.8x sin 0.3x
APPLICATIONS
69. Analytic Geometry. Use the information in the figure to
show that
m2 m1
tan (2 1) 1 m1m2
L2
72. Light Refraction. Use the result of Problem 71 to find to
the nearest degree if 43°, M 0.25 inch, and N 0.11
inch.
2 1
73. 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
Hh
L1
(B) Using sum or difference identities, show that the result
in part A can be written in the form
2
1
1 tan cot 1 tan cot Hh
tan 1 Slope of L1 m1
tan 2 Slope of L2 m2
70. Analytic Geometry. Find the acute angle of intersection
between the two lines y 3x 1 and y 12 x 1. (Use the
results of Problem 69.)
★★ 71. 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
sin ( )
sin ( )
(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 3 significant digits.
El Capitan
D
M
H
Air
E
E
Air
h
Plate
glass
A
N
B
Merced River
C
Yosemite
National Park