Thomas Calculus 13th [Solutions]

22 Chapter 1 Functions
40. sin(2 x) sin 2 cos( x) cos(2 )sin( x) (0)(cos( x)) (1)(sin( x)) sin x
41. sin 3 x sin 3 cos( x) cos 3 sin( x) ( 1)(cos x) (0)(sin( x)) cos x
2 2 2
42. cos 3 x cos 3 cos x sin 3 sin x (0)(cos x) ( 1)(sin x) sin x
2 2 2
43.
44.
45.
46.
7 2 1 2 3 6 2
sin sin sin cos cos sin
12 4 3 4 3 4 3 2 2 2 2 4
11 2 2 2 2 1 2 3 2 6
cos cos cos cos sin sin
12 4 3 4 3 4 3 2 2 2 2 4
1 2 3 2 1 3
cos cos cos cos sin sin
12 3 4 3 4 3 4 2 2 2 2 2 2
5 2 2 2 3 2 1 2 1 3
sin sin sin cos cos sin
12 3 4 3 4 3 4 2 2 2 2 2 2
47.
2 2
8 2
2
1 cos 1 2 2
cos 48.
8 2 2 4
3
2
10 1
2
1 cos
5
12
2 3
cos
12 2 2 4
49.
51.
2 3
12 2
2
1 cos 1 2 3
sin 50.
12 2 2 4
2 3 3
sin sin , 2 , 4 , 5
4 2 3 3 3 3
2
2
6 1
2
1 cos
3
8
2 2
sin
8 2 2 4
52.
2 2
sin 2 cos 2 sin cos tan 2
1 tan 1 3 5 7
2 2
cos cos
4 , 4 , 4 , 4
53. sin 2 cos 0 2sin cos cos 0 cos (2sin 1) 0 cos 0 or 2sin 1 0
cos 0 or sin 1 , 3 , or , 5 , , 5 , 3
2 2 2 6 6 6 2 6 2
54.
2
2
cos 2 cos 0 2cos 1 cos 0 2cos cos 1 0 (cos 1)(2cos 1) 0
cos 1 0 or 2cos 1 0 cos 1 or cos 1 or , 5 , , 5
2
3 3 3 3
55.
56.
tan( A B)
tan( A B)
sin Acos B cos Asin
B
cos Acos B cos Acos
B
cos Acos B sin Asin
B
cos Acos B cos Acos
B
sin( A B) sin Acos B cos Acos B tan A tan B
cos( A B) cos Acos B sin Asin B 1 tan Atan
B
sin Acos B cos Asin
B
cos Acos B cos Acos
B
cos Acos B sin Asin
B
cos Acos B cos Acos
B
sin( A B) sin Acos B cos Acos B tan A tan B
cos( A B) cos Acos B sin Asin B 1 tan Atan
B
2 2 2
57. According to the figure in the text, we have the following: By the law of cosines, c a b 2ab
cos
2 2
1 1 2cos( A B) 2 2cos( A B ) . By distance formula,
2 2 2
c (cos A cos B) (sin A sin B)
2 2 2 2
cos A 2cos Acos B cos B sin A 2sin Asin B sin B 2 2(cos Acos B sin Asin B ) . Thus
2
c 2 2cos( A B) 2 2(cos Acos B sin Asin B) cos( A B) cos Acos B sin Asin
B .
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