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Lesson 7.5 Exercises, pages 641–649
A
3. Verify each identity for the given values of a and b.
a) sin (a ⫹ b) ⫽ sin a cos b ⫹ cos a sin b;
for a ⫽ 90° and b ⫽ 30°
Substitute: A 90° and B 30°
L.S. sin (A B)
sin (90° 30°)
sin 120°
R.S. sin A cos B cos A sin B
sin 90° cos 30° cos 90° sin 30°
√
√
(1)a
3
2
√
3
1
b (0)a b
2
2
3
0
2
√
3
2
The left side is equal to the right side, so the identity is verified.
␲
␲
b) cos (a ⫺ b) ⫽ cos a cos b ⫹ sin a sin b; for a ⫽ 2 and b ⫽ 4
and B 2
4
Substitute: A L.S. cos (A b)
R.S. cos A cos B sin A sin B
sin a b
2
sin
4
cos
cos sin sin
2
4
2
4
(0) a√ b (1) a√ b
1
2
4
1
2
1
2
1
2
√
√
The left side is equal to the right side, so the identity is verified.
c) tan (a ⫹ b) ⫽
Substitute: A tan a + tan b
2␲
␲
; for a ⫽ 3 and b ⫽ 3
1 - tan a tan b
2
and B 3
3
L.S. tan (A B)
2
b
3
3
tan a
tan 0
R.S. tan A tan B
1 tan A tan B
2
tan
3
3
2
tan
1 tan
3
3
√
√
3 3
√ √
1 ( 3)( 3 )
tan
0
The left side is equal to the right side, so the identity is verified.
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4. Simplify each expression.
a) cos 8u cos 2u ⫹ sin 8u sin 2u
cos (8U 2U)
cos 6U
c)
b) cos u sin 4u ⫺ sin u cos 4u
sin 4U cos U cos 4U sin U
sin (4U U)
sin 3U
tan 7x - tan 3x
1 + tan 7x tan 3x
d) sin 5x sin 3x ⫺ cos 5x cos 3x
tan (7x 3x)
tan 4x
(cos 5x cos 3x sin 5x sin 3x)
cos (5x 3x)
cos 8x
B
5. Simplify each expression, then determine its exact value.
␲
␲
a) cos 75° cos 15° ⫺ sin 75° sin 15° b) sin ␲ cos 4 ⫹ cos ␲ sin 4
4
sin a b
cos (75° 15°)
cos 90°
0
5
4
1
√
2
sin
␲
␲
␲
␲
c) cos 3 sin 6 ⫺ sin 3 cos 6
asin
cos cos sin b
3
6
3
6
sin
tan a b
3
4
sin a b
3
␲
3
d)
␲
1 - tan ␲ tan
3
tan ␲ + tan
tan
3
√
3
6
6
1
2
␲
␲
6. a) Expand sin a2 + 2 b to verify that sin ␲ ⫽ 0.
sin a b sin
2
2
cos cos sin
2
2
2
2
sin (1)(0) (0)(1)
0
␲
␲
b) Expand cos a2 + 2 b to verify that cos ␲ ⫽ ⫺1.
cos a b cos
2
2
cos sin sin
2
2
2
2
cos (0)(0) (1)(1)
1
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√
3
␲
␲
␲
c) Expand cos a2 - 3 b to verify that cos 6 ⫽ 2 .
cos a b cos
2
cos sin sin
2
3
2
3
√
3
1
cos (0)a b (1)a b
6
2
2
√
3
2
3
␲
␲
␲
d) Expand tan a6 + 6 b to verify that tan 3 ⫽
tan a b 6
6
tan
3
√
3.
tan
6
6
1 tan tan
6
6
1
1
√ √
3
3
tan
1
1
1 a√ b a√ b
3
3
2
√
√
3
, or 3
2
3
7. Determine each exact value.
a) cos 75°
b) sin 15°
cos (45º 30º)
cos 45º cos 30º sin 45º sin 30º
√
3
1
1
a√ b a b a√ b a b
2
2
2 2
√
31
√
2 2
1
␲
√
3
1
1
1
a√ b a b a√ b a b
2
2
2 2
√
31
√
2 2
5␲
c) tan 12
d) tan 12
tan a b
tan a b
tan
3
4
1 tan tan
3
4
√
31
√
1 ( 3 )(1)
√
31
√
31
tan
6
4
1 tan tan
6
4
1
√ 1
3
3
tan
34
sin (45º 30º)
sin 45º cos 30º cos 45º sin 30º
4
6
4
tan
1
1 a√ b(1)
3
√
1 3
√
31
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1
2
8. a) Given sin b ⫽ - 3 and cos a ⫽ 5 , where angle b is in standard
position with its terminal arm in Quadrant 3 and angle a is in
standard position with its terminal arm in Quadrant 4; determine
each exact value.
i) sin (a ⫺ b)
ii) cos (a ⫺ b)
iii) tan (a ⫹ b)
iv) tan (a ⫺ b)
Use: x2 y2 r2
For angle A,
substitute: x 2, r 5
22 y2 52
√
y 21 since the terminal arm
of angle A lies in Quadrant 4.
√
So, sin A 21
5
Substitute for A and B in:
i) sin (A B)
sin A cos B cos A sin B
√
√
21
8
2
1
a
b a b a b a b
5
3
5
3
√
168
2
15
15
√
168 2
15
For angle B,
substitute: y 1, r 3
x2 (1)2 32
√
x 8 since the terminal arm of
angle B lies in Quadrant 3.
√
So, cos B 8
3
ii) cos (A B)
cos A cos B sin A sin B
√
√
8
21
2
1
a b a b a
b a b
5
3
5
3
√
√
21
2 8
15
15
√
√
2 8 21
15
sin (A B)
tan A tan B
iv) tan (A B) 1 tan A tan B
cos (A B)
√
√
21
1
168 2
√
√
√
2
8
2
8 21
√
21 1
1 a
b a√ b
2
8
√
168 2
√
√
2 8 21
iii) tan (A B) b) What other strategy could you use to determine tan (a ⫺ b)?
Sample response: I could substitute for tan A and tan B in the identity
for tan (A B).
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9. Prove each identity.
␲
a) cos u ⫽ sin a2 + ub
b) ⫺tan u ⫽ tan (␲ ⫺ u)
Use: sin (A B)
sin A cos B cos A sin B
2
Substitute: A , B U
R.S. sin
Quit
cos U cos sin U
2
2
(1) cos U (0) sin U
cos U
L.S.
The left side is equal to the right
side, so the identity is proved.
Use: tan (A B) tan A tan B
1 tan A tan B
Substitute: A , B U
tan tan U
1 tan tan U
0 tan U
1 (0) tan U
R.S. tan U
L.S.
The left side is equal to the right
side, so the identity is proved.
10. For each identity below:
y
i) Use the diagram at the right to
explain why each identity is true.
ii) Prove the identity.
a) cos (⫺u) ⫽ cos u
␪
O
x
⫺␪
i) The cosine of an angle whose terminal
arm lies in Quadrant 4 is positive and
equal to the cosine of its reference
angle in Quadrant 1.
ii) L.S. cos (U)
cos (0 U)
cos 0 cos U sin 0 sin U
(1) cos U (0) sin U
cos U
R.S.
The left side is equal to the right side, so the identity is proved.
b) sin (⫺u) ⫽ ⫺sin u
i) The sine of an angle whose terminal arm lies in Quadrant 4 is
negative and has the same numerical value as the sine of its
reference angle in Quadrant 1.
ii) L.S. sin (U)
sin (0 U)
sin 0 cos U cos 0 sin U
(0) cos U (1) sin U
sin U
R.S.
The left side is equal to the right side, so the identity is proved.
36
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c) tan (⫺u) ⫽ ⫺tan u
i) The tangent of an angle whose terminal arm lies in Quadrant 4 is
negative and has the same numerical value as the tangent of its
reference angle in Quadrant 1.
ii) L.S. tan (U)
sin ( U)
cos ( U)
sin U
cos U
tan U
R.S.
The left side is equal to the right side, so the identity is proved.
11. If f(x) is an even function, then f(⫺x) ⫽ f(x)
If f(x) is an odd function, then f(⫺x) ⫽ ⫺f(x)
Use the identities in question 10 to classify each of the sine, cosine,
and tangent functions as odd or even.
Since sin (x) sin x, the sine function is an odd function
Since cos (x) cos x, the cosine function is an even function
Since tan (x) tan x, the tangent function is an odd function
12. Solve this equation over the domain ⫺90° < x < 270°:
√
tan 4x - tan 3x
⫽ 3
1 + tan 4x tan 3x
√
tan (4x 3x) 3
√
tan x 3
x 60º or x 240º
4
5
13. Given tan a ⫽ 3 and tan b ⫽ - 12 , where angle a is in standard
position in Quadrant 1 and angle b is in standard position with
its terminal arm in Quadrant 2, determine the exact value of
cos (a ⫺ b).
Use: x2 y2 r2
For angle A, substitute:
x 3, y 4
32 42 r2
r5
So, sin A 3
4
and cos A 5
5
For angle B, substitute:
x 12, y 5
(12)2 52 r2
r 13
So, sin B 12
5
and cos B 13
13
Substitute for A and B in:
cos (A B) cos A cos B sin A sin B
a b a
3
5
12
4 5
b a ba b
13
5 13
20
36
65
65
16
65
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14. Prove each identity.
√
␲
␲
a) sin a4 + ub + sin a4 - ub ⫽ 2 cos u
cos U cos sin U sin cos U cos sin U
4
4
4
4
2 sin cos U
4
L.S. sin
2 a√ b cos U
1
2
√
2 cos U
R.S.
The left side is equal to the right side, so the identity is proved.
␲
␲
b) cos a6 + ub ⫺ cos a6 - ub ⫽ ⫺sin u
cos U sin sin U acos cos U sin sin Ub
6
6
6
6
2 sin sin U
6
L.S. cos
2 a b sin U
1
2
sin U
R.S.
The left side is equal to the right side, so the identity is proved.
c) tan (a ⫺ b) ⫽
tan a - tan b
1 + tan a tan b
L.S. tan (A B)
sin (A B)
sin A cos B cos A sin B
cos A cos B sin A sin B
sin A cos B
cos A cos B
cos A cos B
cos A cos B
Expand.
cos (A B)
cos A sin B
cos A cos B
sin A sin B
Simplify.
cos A cos B
sin B
sin A
cos A
cos B
1
Divide numerator and denominator
by cos A cos B.
sin A sin B
Use the tangent identity.
cos A cos B
tan A tan B
1 tan A tan B
R.S.
The left side is equal to the right side, so the identity is proved.
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3␲
␲
15. Solve each equation over the domain - 2 ≤ x < 2 .
a) sin 7x cos 5x ⫺ cos 7x sin 5x ⫽ ⫺1
Consider 3
2
x 4
◊
sin (7x 5x) 1
sin 2x 1
2x<.
5
2
5
x 4
5
The roots are: x and x 4
4
2x or
2x b) cos 2x cos x ⫺ sin 2x sin x ⫽ 0
cos (2x x) 0
cos 3x 0
9
◊ 3x<3
.
2
2
3
5
7
9
3x or or or or or 2
2
2
2
2
2
5
7
3
x or or or or or 6
6
2
6
6
2
5
7
3
The roots are: x — , x , x , x , x 6
2
6
6
2
Consider 1
16. Here are two solutions for solving the equation sin (␲ ⫹ x) ⫽ √
2
over the set of real numbers. Are both solutions correct? Explain.
Solution 1
Solution 2
1
sin (␲ ⫹ x) ⫽ √
2
␲
Either ␲ ⫹ x ⫽ 4 ⫹ 2␲k, k H 3␲
x ⫽ - 4 ⫹ 2␲k, k H 3␲
Or ␲ ⫹ x ⫽ 4 ⫹ 2␲k, k H ␲
x ⫽ - 4 ⫹ 2␲k, k H 1
2
1
sin ␲ cos x ⫹ cos ␲ sin x ⫽ √
2
1
(0) cos x ⫹ (⫺1) sin x ⫽ √
2
1
⫺sin x ⫽ √
2
1
sin x ⫽ ⫺√
2
5␲
x ⫽ 4 ⫹ 2␲k, k H 7␲
or x ⫽ 4 ⫹ 2␲k, k H sin (␲ ⫹ x) ⫽ √
3
2k, k ç becomes
4
7
5
x
, and 2k, k ç becomes .
4
4
4
Both solutions are correct. For k 1, ©P DO NOT COPY.
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17. Determine the general solution of each equation over the set of real
numbers.
1
a) cos (␲ ⫹ x) ⫽ 2
b) tan (␲ ⫹ x) ⫽ ⫺1
1
2
1
(1) cos x (0) sin x 2
cos cos x sin sin x cos x 2
2k, k ç or
3
4
x
2k, k ç 3
1
2
x
tan tan x
1
1 tan tan x
0 tan x
1
1 (0) tan x
tan x 1
3
x
k, k ç 4
C
18. Use algebra to determine the amplitude and the period of the graph
␲
␲
of y ⫽ sin a4 + xb ⫹ sin a4 - xb . Describe your strategy.
Use graphing technology to check.
4
4
y sin a xb sin a xb
Use the sum and difference formulas.
cos x cos sin x sin cos x cos sin x
4
4
4
4
y 2 sin cos x
4
y sin
y 2 a√ b cos x
y
√
1
2
2 cos x
The amplitude is
40
√
2 and the period is 2.
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