Thomas Calculus 13th [Solutions]

Section 1.3 Trigonometric Functions 21
(b) Sin x and csc x are positive for x in the intervals
3 ,
2
and (0, ); and sin x and csc x are
negative for x in the intervals ( , 0) and
, 3
2
. Csc x is undefined when sin x is 0. The
range of csc x is ( , 1] [1, ); the range of
sin x is [ 1,1].
28. Since cot x 1 , cot x is undefined when tan x 0
tan x
and is zero when tan x is undefined. As tan x
approaches zero through positive values, cot x
approaches infinity. Also, cot x approaches negative
infinity as tan x approaches zero through negative
values.
29. D: x ; R: y 1, 0, 1
30. D: x ; R: y 1, 0, 1
31.
32.
33.
34.
cos x cos x cos sin xsin (cos x)(0) (sin x)( 1) sin x
2 2 2
cos x cos x cos sin xsin (cos x)(0) (sin x)(1) sin x
2 2 2
sin x sin x cos cos xsin (sin x)(0) (cos x)(1) cos x
2 2 2
sin x sin x cos cos xsin (sin x)(0) (cos x)( 1) cos x
2 2 2
35. cos( A B) cos( A ( B)) cos Acos( B) sin Asin( B) cos Acos B sin A( sin B)
cos Acos B sin Asin
B
36. sin( A B) sin( A ( B)) sin Acos( B) cos Asin( B) sin Acos B cos A( sin B)
sin Acos B cos Asin
B
37. If B A, A B 0 cos( A B ) cos 0 1. Also cos( A B) cos( A A ) cos A cos A sin A sin A
2 2
2 2
cos A sin A . Therefore, cos A sin A 1.
38. If B 2 , then cos( A 2 ) cos A cos 2 sin Asin 2 (cos A )(1) (sin A)(0) cos A and
sin( A 2 ) sin Acos 2 cos Asin 2 (sin A)(1) (cos A)(0) sin A . The result agrees with the fact that the
cosine and sine functions have period 2 .
39. cos( x) cos cos x sin sin x ( 1)(cos x) (0)(sin x) cos x
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