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sin2 B, given: B) 1. Ans. 45 Ans. 135
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Find the values of sin 2 , cos 2 , and tan 2 , given: (a) sin (b) sin (c) sin (d) tan (e) tan u, 3/5, 3/5, 1/2, 1/5, in quadrant I in quadrant II in quadrant IV in quadrant II Ans. 24/25, 7/25, 24/7 Ans. Ans. Ans. Ans. 1 24/25, 7/25, 23>2, 1>2, 5>13, 12>13, 2u 1 , u2 1 u2 24/7 23 5>12 2u u2
in quadrant I
, u2 1
CHAPTER 9 Trigonometric Functions of Two Angles
9.38 Prove: (a) tan sin 2 2 sin2 1 cos 2 1
(e) cos 2u
(b) cot sin 2 (c) sin3 x
1 1 1 cos 2u (f) sin 2u (g) cos 3
tan2 u tan2 u cot u 3 cos
1 8 cos 4x
cos3 x
sin x cos x 1 sin 2A 1 (d) cos 2A 1 9.39
sin 2x
4 cos3
tan A tan A
(h) cos4 x
1 2 cos 2x
Find the values of the sine, cosine, and tangent of (a) 30 , given cos 60 (b) 105 , given cos 210 (c)
1 2 u,
1/2 23>2
Ans. 1>2, 23>2, 1> 23 Ans.
1 2 #2
23>3 23, (2 23)
1 2 #2
given sin u
3/5, u in quadrant I 7>24, 2u in quadrant I 5/12, 2u in quadrant II
Ans. 1> 210
210>10, 3> 210
3 210>10, 1>3
(d) u, given cot 2u (e) u, given cot 2u 9.40
Ans. 3/5, 4/5, 3/4 Ans. 3> 213 3 213>13, 2> 213 2 213>13, 3>2
Find the values of the sine, cosine, and tangent of (a) 7 /8, given cos 7 /4 (b) 5 /8, given sin 5 /4 22>2 22/2 Ans. 1 #2 2 Ans.
1 2 #2
22, 22,
1 2 #2 1 2 #2
22, #3 22, #3
2 22 2 22
Prove: (a) cos x (b) sin x (c) A sin 1 u 2 (d) tan (e) (f) 1 1 1
1 2u
2 cos2 2 x
1 2 sin2 2 x
1 1 2 sin 2 x cos 2 x
cos 1 u R 2 2 csc u
1 2u 1 2u
1 cot u
sin u
tan tan
sin u cos u
cos u sin u
2 tan 1 x 2
1 tan2 2 x
In the right triangle ABC in which C is the right angle, prove: sin 2A sin 3x sin x C 2ab c2 cos 2A b2 c2 a2 sin 1 A 2 c b A 2c 2 tan 10 . cos 1 A 2 c b A 2c
9.43 9.44
Prove (a) If A B
cos 3x cos x 180 , prove: sin C cos C sin2 C tan
1 2B
2 and (b) tan 50
tan 40
(a) sin A (b) cos A (c) sin2 A (d) tan
1 2A
sin B cos B sin2 B tan
1 2B
4 cos 1 A cos 1 B cos 1 C 2 2 2 1 4 sin 1 A sin 1 B sin 1 C 2 2 2
2 sin A sin B cos C tan 1 C tan 1 A 2 2 1
tan 1 C 2
Sum, Difference, and Product Formulas
10.1 Products of Sines and Cosines
sin cos cos sin cos sin cos sin
1 2 [sin 1 2 [sin 1 2
) ) ) ( )
sin ( sin ( cos ( cos (
)] )] )] )]
[cos (
1 2 [cos
For proofs of these formulas, see Prob. 10.1.
10.2 Sum and Difference of Sines and Cosines
sin A sin A cos A cos A sin B sin B cos B cos B 2 sin 1(A 2 2 cos 2(A 2 cos 2(A 2 sin 2 (A
1 1 1
B) cos 2(A B) sin 2(A B) cos 2(A B) sin 2 (A
1 1 1
B) B) B) B)
For proofs of these formulas, see Prob. 10.2.
SOLVED PROBLEMS
10.1 Derive the product formulas.
Since sin ( ) sin ( sin Since Since sin ( cos ( ) ) sin ( cos cos ( cos cos sin ) cos ) ) (sin 2 sin
cos cos ) sin , ) cos cos )
cos ) )] )]
(sin
sin )
[sin ( [sin (
sin ( sin ( sin sin )
2 cos
(cos 2 cos
(cos )]
sin )
[cos (
cos (
CHAPTER 10 Sum, Difference, and Product Formulas
Since
cos (
cos ( sin sin
2 sin
sin ) cos ( )]
[cos (
10.2 Derive the sum and difference formulas.
Let sin ( sin ( cos ( cos ( ) ) ) ) A and sin ( sin ( cos ( cos ( ) ) ) ) B so that 2 sin 2 cos 2 cos 2 sin cos sin cos cos
B) and
1 2 (A
B). Then (see Prob. 10.1) sin B sin B cos B cos B 2 sin 2 (A
1 2 cos 2(A 2 cos 1(A 2 1 2 sin 2 (A 1
becomes becomes becomes becomes
sin A sin A cos A cos A
B) cos 2 (A B) sin 1(A 2 B) cos 1(A 2 1 B) sin 2 (A
B) B) B) B)
10.3 Express each of the following as a sum or difference. (a) sin 40 cos 30 ,
(a) sin 40 cos 30 (b) cos 110 sin 55 (c) cos 50 cos 35 (d) sin 55 sin 40
(b) cos 110 sin 55 ,
1 2 [sin (40 1 2 [sin (110 1 [cos (50 2 1 2 [cos (55
(c) cos 50 cos 35 ,
30 )] 55 35 )] 40
(d) sin 55 sin 40
30 ) 55 ) 35 )
sin (40 sin (110 cos (50 cos (55
40 )
1 sin 10 ) 2 (sin 70 1 )] 2(sin 165 sin 55 ) 1 (cos 85 cos 15 ) 2 1 )] cos 15 2 (cos 95
10.4 Express each of the following as a product. (a) sin 50
(a) sin 50 (b) sin 70 (c) cos 55 (d) cos 35
sin 40 , (b) sin 70
sin 40 sin 20 cos 25 cos 75
1 2 sin 2(50
sin 20 , (c) cos 55
1 40 ) cos 2(50
cos 25 , (d) cos 35
cos 75
40 ) 20 ) 25 ) 75 )
2 sin 45 cos 5 2 cos 45 sin 25 2 cos 40 cos 15 2 sin 55 sin ( 20 ) 2 sin 55 sin 20
2 cos 1(70 2 1 2 cos 2(55 2 sin 1 (35 2
20 ) sin 1(70 2 1 25 ) cos 2(55 75 ) sin 1 (35 2
10.5 Prove
sin 4A cos 4A
sin 2A cos 2A
tan 3A.
2 sin 2(4A 2 cos 1(4A 2
sin 4A cos 4A
sin 2A cos 2A
2A) cos 2(4A 2A) cos 1(4A 2
2A) 2A)
sin 3A cos 3A
tan 3A
10.6 Prove
sin A sin A
sin B sin B
sin B sin B
tan 1(A 2 tan 1(A 2
2 cos 2(A 2 sin 1(A 2
1 16 (2
B) B)
B) B) cot 1(A 2 B) tan 1(A 2 tan 2(A tan 1(A 2
sin A sin A
B) sin 1(A 2 B) cos 1(A 2
B) B)
10.7 Prove cos3 x sin2 x
cos x
cos 3x
cos 5x).
1 2 4 sin
cos3 x sin2 x
(sin x cos x)2 cos x
1 4 (sin 1 8{ 1 16 (2 1 2x) [2(sin 3x
2x cos x
1 8 (sin 1 2 (cos
1 4 (sin
2x)(sin 2x cos x) sin 2x sin x)
sin x)] [
3x sin 2x cos x)]}
1 2 (cos
cos x) cos 3x
cos x
cos 5x)
10.8 Prove 1
cos 2x
(cos 2x
cos 4x
cos 4x)
cos 6x
cos 6x 1
4 cos x cos 2x cos 3x.
2 cos 3x cos x cos 6x (1 cos 6x) 2 cos 3x cos x cos x) 2 cos2 3x 2 cos 3x cos x 2 cos 3x (cos 3x 4 cos x cos 2x cos 3x
2 cos 3x (2 cos 2x cos x)
CHAPTER 10 Sum, Difference, and Product Formulas
3 sin x into the form c cos (x
) c(cos x cos 4/5, sin 3/c. Since sin2 3/5, and 3 sin x 4 cos x Using c 5, 3.7851 rad and 4 cos x 3 sin x 5 cos (x 3.7851) cos2 5 cos (x
10.9 Transform 4 cos x
Since c cos (x Then cos Using c 5, cos
4 and c sin 5 and 5. 3. 1, c
sin x sin ), set c cos
4/c and sin
0.6435 rad. Thus, 0.6435).
10.10 Find the maximum and minimum values of 4 cos x
From Prob. 10.9, 4 cos x 3 sin x 5 cos (x
3 sin x on the interval 0
0.6435).
On the prescribed interval, cos attains its maximum value 1 when 0 and its minimum value 1 when . Thus, the maximum value of 4 cos x 3 sin x is 5, which occurs when x 0.6435 0 or when x 0.6435, while the minimum value is 5, which occurs when x 0.6435 or when x 3.7851.
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