Download Sum, Difference, and Cofunction Identities

522
CHAPTER 6
TRIGONOMETRIC IDENTITIES AND EQUATIONS
In Exercises 57 to 64, compare the graphs of each side
of the equation to predict whether the equation is an
identity.
57. sin 2x = 2 sin x cos x
58. sin2 x + cos2 x = 1
59. sin x + cos x = 22 sin a x +
p
b
4
69. tan4 x - sec4 x = tan2 x + sec 2 x
70. 21 + tan2x = sec x
In Exercises 71 to 76, verify the identity.
71.
1 - sin x + cos x
cos x
=
1 + sin x + cos x
sin x + 1
60. cos 2x = 2 cos2 x - 1
72.
1 - tan x + sec x
1 + sec x
=
1 + tan x - sec x
tan x
61. cos ax +
73.
p
p
p
b = cos x cos - sin x sin
3
3
3
p
p
p
+ sin x sin
62. cos ax - b = cos x cos
4
4
4
2 sin2 x
= 1 -
63. sin ax +
p
p
p
b = sin x sin + cos x cos
6
6
6
74.
64. sin ax -
p
p
p
b = sin x cos + cos x sin
3
3
3
75.
In Exercises 65 to 70, verify that the equation is not an
identity by finding an x value for which the left side of
the equation is not equal to the right side.
2 sin4 x + 2 sin2 x cos 2 x - 3 sin2 x - 3 cos2 x
76.
3
csc 2 x
2
4 tan x sec 2 x - 4 tan x - sec 2 x + 1
4 tan3 x - tan2 x
sin x1tan x + 12 - 2 tan x cos x
sin x - cos x
= tan x
sin2 x cos x + cos3 x - sin3 x cos x - sin x cos3 x
cos x
=
1 + sin x
65. 1sin x + cos x22 = sin2 x + cos2 x
1 - sin2 x
In Exercises 77 and 78, verify the identity by completing
the square of the left side of the identity.
66. tan 2x = 2 tan x
67. cos1x + 30°2 = cos x + cos 30°
77. sin4 x + cos4 x = 1 - 2 sin2 x cos2 x
68. 21 - sin2 x = cos x
78. tan4 x + sec4 x = 1 + 2 tan2 x sec 2 x
SECTION 6.2
b)
Identities That Involve (a
Cofunctions
Additional Sum and Difference
Identities
Reduction Formulas
= 1
Sum, Difference, and Cofunction Identities
PREPARE FOR THIS SECTION
Prepare for this section by completing the following exercises. The answers can be found
on page A41.
PS1. Compare cos1a - b2 and cos a cos b + sin a sin b for a =
p
p
and b = . [5.2]
2
6
PS2. Compare sin1a + b2 and sin a cos b + cos a sin b for a =
p
p
and b = . [5.2]
2
3
PS3. Compare sin190° - u2 and cos u for u = 30°, u = 45°, and u = 120°. [5.2]
PS4. Compare tan a
4p
p
p
p
- ub and cot u for u = , u = , and u =
. [5.2]
2
6
4
3
tan a - tan b
p
p
PS5. Compare tan1a - b2 and
for a =
and b = . [5.2]
1 + tan a tan b
3
6
PS6. Find the value of sin312k + 12p4, where k is any arbitrary integer. [5.2]
6.2
SUM, DIFFERENCE, AND COFUNCTION IDENTITIES
Identities That Involve (a
523
b)
Each identity in Section 6.1 involved only one variable. We now consider identities that
involve a trigonometric function of the sum or difference of two variables.
Sum and Difference Identities
cos1a - b2 = cos a cos b + sin a sin b
cos1a + b2 = cos a cos b - sin a sin b
sin1a - b2 = sin a cos b - cos a sin b
sin1a + b2 = sin a cos b + cos a sin b
tan1a + b2 =
tan a + tan b
1 - tan a tan b
tan1a - b2 =
tan a - tan b
1 + tan a tan b
Proof
To establish the identity for cos1a - b2, we use the unit circle shown in Figure 6.2. The
angles a and b are drawn in standard position, with OA and OB as the terminal sides of a
and b , respectively. The coordinates of A are 1cos a, sin a2, and the coordinates of B are
1cos b, sin b2. The angle 1a - b2 is formed by the terminal sides of the angles a and b
(angle AOB).
y
C(cos (␣ − β ), sin ( ␣ − β ) )
β
B(cos β , sin β )
␣
␣− β
␣−β
O
D(1, 0)
x
A(cos ␣, sin ␣)
Figure 6.2
An angle equal in measure to angle 1a - b2 is placed in standard position in the same
figure (angle COD). From geometry, if two central angles of a circle have the same measure, then their chords are also equal in measure. Thus the chords AB and CD are equal in
length. Using the distance formula, we can calculate the lengths of the chords AB and CD.
d1A, B2 = 21cos a - cos b22 + 1sin a - sin b22
d1C, D2 = 23cos1a - b2 - 142 + 3sin1a - b2 - 042
Because d1A, B2 = d1C, D2, we have
21cos a - cos b22 + 1sin a - sin b22 = 23cos 1a - b2 - 142 + 3sin 1a - b242
524
CHAPTER 6
TRIGONOMETRIC IDENTITIES AND EQUATIONS
Squaring each side of the equation and simplifying, we obtain
1cos a - cos b22 + 1sin a - sin b22 = 3cos1a - b2 - 142 + 3sin1a - b242
cos2 a - 2 cos a cos b + cos2 b + sin2 a - 2 sin a sin b + sin2 b
= cos21a - b2 - 2 cos1a - b2 + 1 + sin21a - b2
cos2 a + sin2 a + cos2 b + sin2 b - 2 cos a cos b - 2 sin a sin b
= cos21a - b2 + sin21a - b2 + 1 - 2 cos1a - b2
Simplifying by using sin2 u + cos2 u = 1, we have
2 - 2 sin a sin b - 2 cos a cos b = 2 - 2 cos1a - b2
Solving for cos1a - b2 gives us
cos1a - b2 = cos a cos b + sin a sin b
◆
To derive an identity for cos1a + b2, write cos1a + b2 as cos3a - 1 - b24.
cos1a + b2 = cos3a - 1 - b24 = cos a cos1- b2 + sin a sin1- b2
Recall that cos1 - b2 = cos b and sin1 - b2 = - sin b . Substituting into the previous
equation, we obtain the identity
cos1a + b2 = cos a cos b - sin a sin b
EXAMPLE 1
Evaluate a Trigonometric Expression
Use an identity to find the exact value of cos160° - 45°2.
Solution
Use the identity cos1a - b2 = cos a cos b + sin a sin b with a = 60° and b = 45°.
cos160° - 45°2 = cos 60° cos 45° + sin 60° sin 45°
1
22
23
22
= a ba
b + a
ba
b
2
2
2
2
=
26
22
+
4
4
=
22 + 26
4
• Substitute.
• Evaluate each factor.
• Simplify.
Try Exercise 4, page 529
Cofunctions
Any pair of trigonometric functions f and g for which f1x2 = g190° - x2 and
g1x2 = f190° - x2 are said to be cofunctions.
6.2
Study tip
To visualize the cofunction identities, consider the right triangle
shown in the following figure.
c
90° − θ
SUM, DIFFERENCE, AND COFUNCTION IDENTITIES
525
Cofunction Identities
sin190° - u2 = cos u
cos190° - u2 = sin u
tan190° - u2 = cot u
cot190° - u2 = tan u
sec190° - u2 = csc u
csc190° - u2 = sec u
b
If u is in radian measure, replace 90° with
θ
a
If u is the degree measure of one
of the acute angles, then the
degree measure of the other
acute angle is 190° - u2. Using
the definitions of the trigonometric functions gives us
b
= cos190° - u2
c
b
= cot190° - u2
tan u =
a
c
= csc190° - u2
sec u =
a
sin u =
These identities state that the
value of a trigonometric function
of u is equal to the cofunction of
the complement of u.
p
.
2
To verify that the sine function and the cosine function are cofunctions, we use the
identity for cos1a - b2.
cos190° - b2 = cos 90° cos b + sin 90° sin b
= 0 # cos b + 1 # sin b
which gives
cos190° - b2 = sin b
Thus the sine of an angle is equal to the cosine of its complement. Using
cos190° - b2 = sin b with b = 90° - a, we have
cos a = cos390° - 190° - a24 = sin190° - a2
Therefore,
cos a = sin190° - a2
We can use the ratio identities to show that the tangent and cotangent functions are
cofunctions.
tan190° - u2 =
cot190° - u2 =
sin190° - u2
cos190° - u2
cos190° - u2
sin190° - u2
=
cos u
= cot u
sin u
=
sin u
= tan u
cos u
The secant and cosecant functions are also cofunctions.
EXAMPLE 2
Write an Equivalent Expression
Use a cofunction identity to write an equivalent expression for sin 20°.
Solution
The value of a given trigonometric function of u, measured in degrees, is equal to its
cofunction of 90° - u. Thus
sin 20° = cos190° - 20°2
= cos 70°
Try Exercise 20, page 529
526
CHAPTER 6
TRIGONOMETRIC IDENTITIES AND EQUATIONS
Additional Sum and Difference Identities
We can use the cofunction identities to verify the remaining sum and difference identities. To derive an identity for sin1a + b2, substitute a + b for u in the cofunction
identity sin u = cos190° - u2.
sin u = cos190° - u2
sin1a + b2 = cos390° - 1a + b24
• Replace u with a + b.
= cos3190° - a2 - b4
• Rewrite as the difference of two angles.
= cos190° - a2 cos b + sin190° - a2 sin b
= sin a cos b + cos a sin b
Therefore,
sin1a + b2 = sin a cos b + cos a sin b
We also can derive an identity for sin1a - b2 by rewriting 1a - b2 as
3a + 1- b24.
sin1a - b2 = sin3a + 1 - b24
= sin a cos1- b2 + cos a sin1- b2
= sin a cos b - cos a sin b
• cos1 - b2 = cos b
sin1 - b2 = - sin b
Thus
sin1a - b2 = sin a cos b - cos a sin b
The identity for tan1a + b2 is a result of the identity tan u =
sin u
and the identities
cos u
for sin1a + b2 and cos1a + b2.
tan1a + b2 =
sin1a + b2
cos1a + b2
=
sin a cos b + cos a sin b
cos a cos b - sin a sin b
cos a sin b
sin a cos b
+
cos a cos b
cos a cos b
=
cos a cos b
sin a sin b
cos a cos b
cos a cos b
• Multiply both the numerator and the
1
denominator by
and
cos a cos b
simplify.
Therefore,
tan1a + b2 =
tan a + tan b
1 - tan a tan b
The tangent function is an odd function, so tan1- u2 = - tan u. Rewriting 1a - b2 as
3a + 1- b24 enables us to derive an identity for tan1a - b2.
tan1a - b2 = tan3a + 1 - b24 =
tan a + tan1 - b2
1 - tan a tan1- b2
Therefore,
tan1a - b2 =
tan a - tan b
1 + tan a tan b
6.2
SUM, DIFFERENCE, AND COFUNCTION IDENTITIES
527
The sum and difference identities can be used to simplify some trigonometric
expressions.
EXAMPLE 3
Simplify Trigonometric Expressions
Write each expression in terms of a single trigonometric function.
a.
sin 5x cos 3x - cos 5x sin 3x
tan 4a + tan a
1 - tan 4a tan a
b.
Solution
a.
sin 5x cos 3x - cos 5x sin 3x = sin15x - 3x2 = sin 2x
b.
tan 4a + tan a
= tan14a + a2 = tan 5a
1 - tan 4a tan a
Try Exercise 26, page 529
EXAMPLE 4
Given tan a = -
Evaluate a Trigonometric Function
4
5
for a in Quadrant II and tan b = for b in Quadrant IV, find
3
12
sin1a + b2.
Solution
y
4
= - and the terminal side of a is in Quadrant II,
x
3
P11 -3, 42 is a point on the terminal side of a. Similarly, P2112, -52 is a point on the
terminal side of b.
See Figure 6.3. Because tan a =
y
y
4
β
P1(−3, 4)
O
5
6
P
−4
12
x
13
α
O
4
−6
x
P2 (12, −5)
Figure 6.3
Using the Pythagorean Theorem, we find that the length of the line segment OP1 is 5
and the length of OP2 is 13.
sin1a + b2 = sin a cos b + cos a sin b
=
Try Exercise 38, page 530
-3
4 # 12
+
5 13
5
#
63
-5
48
15
=
+
=
13
65
65
65
528
CHAPTER 6
TRIGONOMETRIC IDENTITIES AND EQUATIONS
EXAMPLE 5
Verify an Identity
Verify the identity cos1p - u2 = - cos u.
Solution
cos1p - u2 = cos p cos u + sin p sin u
Plot1 Plot2 Plot3
\Y1 = cos(π–x)
\Y2 = -cos(x)
• Use the identity for
cos1a - b2.
= - 1 # cos u + 0 # sin u
= -cos u
1.5
Try Exercise 50, page 530
−2π
2π
Figure 6.4 shows the graphs of Y1 = cos1p - u2 and Y2 = - cos u on the same
coordinate axes. The fact that the graphs appear to be identical supports the verification in Example 5.
Y1 = Y2
−1.5
Figure 6.4
EXAMPLE 6
Verify an Identity
Verify the identity
sin 4u
cos 5u
cos 4u
=
.
sin u
cos u
sin u cos u
Solution
Subtract the fractions on the left side of the equation.
sin 4u
cos 4u cos u - sin 4u sin u
cos 4u
=
sin u
cos u
sin u cos u
=
=
cos14u + u2
sin u cos u
• Use the identity for cos 1a + b2.
cos 5u
sin u cos u
Try Exercise 62, page 531
cos 4u sin 4u
cos 5u
and the graph of Y2 =
sin u
cos u
sin u cos u
on the same coordinate axes. The fact that the graphs appear to be identical supports the
verification in Example 6.
Figure 6.5 shows the graph of Y1 =
Plot1 Plot2 Plot3
\Y1 = (cos(4x))/sin(x)–(sin(4x)
)/cos(x)
\Y2 = (cos(5x))/(sin(x)cos(x))
7
Reduction Formulas
−2π
2π
Y1 = Y2
−7
Figure 6.5
The sum or difference identities can be used to write expressions such as
sin1u + kp2
sin1u + 2kp2
cos3u + 12k + 12p4
where k is an integer, as expressions involving only sin u or cos u. The resulting formulas
are called reduction formulas.
6.2
EXAMPLE 7
SUM, DIFFERENCE, AND COFUNCTION IDENTITIES
529
Find Reduction Formulas
Write as a function involving only sin u.
sin3u + 12k + 12p4, where k is an integer
Solution
Applying the identity sin1a + b2 = sin a cos b + cos a sin b yields
sin3u + 12k + 12p4 = sin u cos312k + 12p4 + cos u sin312k + 12p4
If k is an integer, then 2k + 1 is an odd integer. The cosine of any odd multiple of
p equals -1, and the sine of any odd multiple of p is 0. This gives us
sin3u + 12k + 12p4 = 1sin u21-12 + 1cos u2102 = - sin u
Thus sin3u + 12k + 12p4 = - sin u for any integer k.
Try Exercise 76, page 531
Question • Is sin1u + 2kp2 = sin u, where k is an integer, a reduction formula?
EXERCISE SET 6.2
In Exercises 1 to 18, find (if possible) the exact value
of the expression.
1. sin145° + 30°2
2. sin1330° + 45°2
16. cos
p
p
p
p
cos - sin
sin
12
4
12
4
p
7p
- tan
12
4
17.
7p
p
1 + tan
tan
12
4
tan
3. cos145° - 30°2
4. cos1120° - 45°2
5. tan145° - 30°2
6. tan1240° - 45°2
7. sin a
8. sin a
5p
p
- b
4
6
3p
p
9. cos a
+ b
4
6
11. tan a
p
p
+ b
6
4
p
p
10. cos a - b
4
3
12. tan a
13. cos 212° cos 122° + sin 212° sin 122°
14. sin 167° cos 107° - cos 167° sin 107°
5p
p
5p
p
15. sin
cos - cos
sin
12
4
12
4
p
4p
+ b
3
4
p
11p
- b
6
4
p
p
+ tan
6
3
18.
p
p
1 - tan tan
6
3
tan
In Exercises 19 to 24, use a cofunction identity to write
an equivalent expression for the given value.
19. sin 42°
20. cos 80°
21. tan 15°
22. cot 2°
23. sec 25°
24. csc 84°
In Exercises 25 to 36, write each expression in terms
of a single trigonometric function.
25. sin 7x cos 2x - cos 7x sin 2x
26. sin x cos 3x + cos x sin 3x
27. cos x cos 2x + sin x sin 2x
Answer • Yes. sin1u + 2kp2 = sin u cos12kp2 + cos u sin12kp2
= 1sin u2112 + 1cos u2102 = sin u
530
CHAPTER 6
TRIGONOMETRIC IDENTITIES AND EQUATIONS
29. sin 7x cos 3x - cos 7x sin 3x
8
7
, a in Quadrant IV, and cos b =
, b
25
17
in Quadrant IV, find
30. cos x cos 5x - sin x sin 5x
a. sin1a + b2
28. cos 4x cos 2x - sin 4x sin 2x
42. Given sin a = -
31. cos 4x cos1-2x2 - sin 4x sin1-2x2
3
15
, a in Quadrant I, and sin b = - , b in
17
5
Quadrant III, find
a. sin1a + b2
2x
x
2x
x
cos
+ cos sin
3
3
3
3
a. sin1a + b2
a. sin1a - b2
15
4
37. Given tan a = - , a in Quadrant II, and tan b =
, b
3
8
in Quadrant III, find
b. cos1a + b2
b. cos1a + b2
24
4
, a in Quadrant II, and cos b = - , b in
25
5
Quadrant III, find
b. cos1a + b2
a. sin1a - b2
b. cos1a + b2
c. tan1a + b2
5
3
, a in Quadrant I, and tan b =
, b in
5
12
Quadrant III, find
47. Given sin a =
a. sin1a + b2
b. cos1a - b2
c. tan1a - b2
15
7
, a in Quadrant I, and tan b = - , b in
8
24
Quadrant IV, find
48. Given tan a =
a. sin1a - b2
b. cos1a - b2
c. tan1a + b2
In Exercises 49 to 74, verify the identity.
49. cos a
p
- ub = sin u
2
50. cos1u + p2 = - cos u
c. tan1a + b2
12
4
41. Given sin a = - , a in Quadrant III, and cos b = ,
5
13
b in Quadrant II, find
a. sin1a - b2
8
24
, a in Quadrant IV, and sin b = , b
17
25
in Quadrant III, find
46. Given cos a =
c. tan1a - b2
40. Given sin a =
b. sin1a + b2
c. tan1a + b2
c. tan1a - b2
3
5
39. Given sin a = , a in Quadrant I, and cos b = - , b in
5
13
Quadrant II, find
b. cos1a + b2
b. cos1a + b2
c. tan1a - b2
8
24
38. Given tan a =
, a in Quadrant I, and sin b = - , b in
7
17
Quadrant III, find
a. cos1b - a2
c. tan1a - b2
5
3
, a in Quadrant III, and sin b =
, b
5
13
in Quadrant I, find
In Exercises 37 to 48, find the exact value of the given
functions.
a. sin1a - b2
b. cos1a + b2
45. Given cos a = -
tan 2x - tan 3x
36.
1 + tan 2x tan 3x
a. sin1a + b2
c. tan1a - b2
12
7
, a in Quadrant II, and sin b = ,b
25
13
in Quadrant IV, find
tan 3x + tan 4x
1 - tan 3x tan 4x
a. sin1a - b2
b. cos1a - b2
44. Given cos a = -
x
3x
x
3x
34. cos
cos + sin
sin
4
4
4
4
35.
c. tan1a + b2
43. Given cos a =
32. sin1 - x2 cos 3x - cos1 - x2 sin 3x
33. sin
b. cos1a - b2
c. tan1a + b2
51. sin a u +
p
b = cos u
2
53. tan au +
p
2 tan u
tan u + 1
54. tan 2u =
b =
4
1 - tan u
1 - tan2 u
52. sin1u + p2 = - sin u
6.2
55. cos a
57. cot a
3p
- u b = - sin u
2
p
- u b = tan u
2
56. sin a
58. cot1p + u2 = cot u
60. sec a
59. csc1p - u2 = csc u
3p
+ ub = - cos u
2
p
- u b = csc u
2
61. sin 6x cos 2x - cos 6x sin 2x = 2 sin 2x cos 2x
SUM, DIFFERENCE, AND COFUNCTION IDENTITIES
77. tan1u + p2
78. cos3u + 12k + 12p4
79. sin1u + 2kp2
80. sin1u - kp2
531
In Exercises 81 to 84, compare the graphs of each side
of the equation to predict whether the equation is an
identity.
81. sin a
p
- xb = cos x
2
82. cos1x + p2 = - cos x
83. sin 7x cos 2x - cos 7x sin 2x = sin 5x
62. cos 5x cos 3x + sin 5x sin 3x = cos2 x - sin2 x
84. sin 3x = 3 sin x - 4 sin3 x
In Exercises 85 to 89, verify the identity.
63. cos1a + b2 + cos1a - b2 = 2 cos a cos b
85. sin1x - y2 # sin1x + y2 = sin2 x cos2 y - cos2 x sin2 y
64. cos1a - b2 - cos1a + b2 = 2 sin a sin b
86. sin1x + y + z2 = sin x cos y cos z + cos x sin y cos z +
cos x cos y sin z - sin x sin y sin z
65. sin1a + b2 + sin1a - b2 = 2 sin a cos b
87. cos1x + y + z2 = cos x cos y cos z - sin x sin y cos z -
66. sin1a - b2 - sin1a + b2 = - 2 cos a sin b
67.
68.
69.
70.
cos1a - b2
=
cot a + tan b
1 + cot a tan b
88.
=
1 + cot a tan b
1 - cot a tan b
89.
sin1a + b2
sin1a + b2
sin1a - b2
sin1x + h2 - sin x
h
cos1x + h2 - cos x
71. sin a
sin x cos y sin z - cos x sin y sin z
h
sin h
+ sin x
h
#
= cos x
= cos x
#
1cos h - 12
h
#
1cos h - 12
h
- sin x
#
p
+ a - b b = cos a cos b + sin a sin b
2
72. cos a
p
+ a + b b = - 1sin a cos b + cos a sin b2
2
73. sin 3x = 3 sin x - 4 sin3 x
sin h
h
sin1x + y2
sin x sin y
cos1x - y2
cos x sin y
= cot x + cot y
= cot y + tan x
90. Model Resistance The drag (resistance) on a fish when it is
swimming is two to three times the drag when it is gliding. To
compensate for this, some fish swim in a saw-tooth pattern, as
shown in the accompanying figure. The ratio of the amount of
energy the fish expends when swimming upward at angle b
and then gliding down at angle a to the energy it expends
swimming horizontally is given by
ER =
where k is a value such that 2 … k … 3, and k depends
on the assumptions we make about the amount of drag experienced by the fish. Find ER for k = 2, a = 10°, and b = 20°.
74. cos 3x = 4 cos3 x - 3 cos x
In Exercises 75 to 80, write the given expression as
a function that involves only sin U, cos U, or tan U.
(In Exercises 78 to 80, assume k is an integer.)
75. cos1u + 3p2
76. sin1u + 2p2
k sin a + sin b
k sin1a + b2
α
β