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10.11 Express each of the following products as a sum or difference of sines or cosines. (a) sin 35 cos 25 (b) sin 25 cos 75 (c) cos 50 cos 70 (d) sin 130 sin 55 (e) sin 4x cos 2x (f) sin x/2 cos 3x/2 (g) cos 7x cos 4x (h) sin 5x sin 4x 10.12 Show that (a) 2 sin 45 cos 15 (b) 2 sin 82 1 cos 37 1 2 2 (c) 2 sin 127 1 sin 97 1 2 2 23 2 23 2 23 2 22 1 and cos 15 22 26 4 22
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60 100 120
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sin 10 ) sin 50 ) cos 20 ) cos 75 )
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sin 2x) sin x) cos 3x) cos x)
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10.13 Express each of the following as a product. (a) (b) (c) (d) (e) (f) (g) (h) sin 50 sin 75 cos 65 cos 80 sin 4x sin 7 cos 6 cos 3x/2 sin 20 2 sin 35 cos 15 sin 35 2 cos 55 sin 20 cos 15 2 cos 40 cos 25 cos 70 2 sin 75 sin 5 sin 2x 2 sin 3x cos x sin 3 2 cos 5 sin 2 cos 2 2 cos 4 cos 2 cos 9x/2 2 sin 3x sin 3x/2
CHAPTER 10 Sum, Difference, and Product Formulas
10.14 Show that (a) sin 40 (b) sin 105 (c) cos 465 (d) sin 75 cos 75 sin 20 sin 15 cos 165 sin 15 cos 15 cos 10 26>2 26>2 23>3
10.15 Prove: (a) (b) (c) (d) (e) (f) (g) sin A cos A sin 2A cos 2A sin A sin A sin 3A cos 3A sin 4A cos 4A sin B sin B tan 2A tan 3A tan 2(A tan 1(A 2
B) cos A cos B 1 cot 2(A B) cot 1(A B) 2 cos A cos B sin sin 2 sin 3 sin 2 (sin sin 3 ) sin 2 (1 2 cos ) cos cos 2 cos 3 cos 2 (1 2 cos ) sin 2 sin 4 sin 6 (sin 2 sin 4 ) 2 sin 3 cos 3 4 cos cos 2 sin 3 ( sin 3x sin 9x) ( sin 5x sin 7x) sin 3x sin 5x sin 7x sin 9x cos 3x cos 5x cos 7x cos 9x ( cos 3x cos 9x) ( cos 5x cos 7x)
tan 6x
10.16 Prove: (a) cos 130 10.17 Prove: (a) cos2 (b) cos
cos 110
cos 10
(b) cos 220
cos 100
cos 20
sin3 sin
1 16 (2 1 32 (2
sin cos 2
sin 3
sin 5 ) 2 cos 4 cos 5 ) sin 5 ) cos 6 )
(c) cos5 (d) sin
1 16 (10 1 16 (10
cos sin
5 cos 3 5 sin 3
10.18 Transform (using radians): (a) 4 cos x (b) 4 cos x (c) sin x (d) 5 cos 3t 3 sin x into the form c sin (x 3 sin x into the form c sin (x cos x into the form c sin (x ) ) ) ) Ans. 5 sin (x Ans. 5 sin (x Ans. 22 sin (x Ans. 13 cos (3t 0.9273) 5.3559) /4) 1.1760)
12 sin 3t into the form c cos (3t
10.19 Find the maximum and minimum values of each sum of Prob. 10.18 and a value of x or t between 0 and 2 at which each occurs. Ans. (a) Maximum 5, when x (b) Same as (a). (c) Maximum (d) Maximum 13, when t 0.6435 (i.e., when x 3 /4; minimum 0.3920; minimum 0.9273 /2); minimum 7 /4. 1.4392. 5, when x 3.7851.
22, when x
22, when x 13, when t
Oblique Triangles
11.1 Oblique Triangles
An oblique triangle is one which does not contain a right angle. Such a triangle contains either three acute angles or two acute angles and one obtuse angle. The convention of denoting the angles by A, B, and C and the lengths of the corresponding opposite sides by a, b, and c will be used here. (See Fig. 11.1.)
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