Page 51 - mathsvol1ch1to3ans

51

15 15
cos 2 cos 2

15 2 2 1 + cos 15
cot = = =
2 15 15 15 sin 15
sin 2 sin cos
2 2 2
√ √ √ √
1 3 1 1 3 + 1 6 + 2
cos 15 = cos(45 − 30) = cos 45 cos 30 + sin 45 sin 30 = √ + √ = √ =
2 2 2 2 √ 2 2 √ 4 √
√
1 3 1 1 3 − 1 6 − 2
sin 15 = sin(45 − 30) = sin 45 cos 30 − cos 45 sin 30 = √ − √ = √ =
2 2 2 2 2 2 4
√ √
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6 + 2
1 + √ √
4
15 1 + cos 15 4 + 6 + 2
cot = = √ √ ! = √ √
2 sin 15 6 − 2 6 − 2
4
√ √ √ √ √ √ √
√ √ √ √ √
15 4 + 6 + 2 6 + 2 4 6 + 6 + 2 3 + 4 2 + 2 3 + 2
cot = = = 6 + 4 + 3 + 2
2 4 4
n
n
10. Prove that (1 + sec 2θ)(1 + sec 4θ) . . . (1 + sec 2 θ) = tan 2 θ cot θ.
√ π π π π π
11. Prove that 32( 3) sin cos cos cos cos = 3.
48 48 24 12 6
Exercise 3.6:
1. Express each of the following as a sum or difference

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(i) sin 35 cos 28 (ii) sin 4x cos 2x (iii) 2 sin 10θ cos 2θ (iv) cos 5θ cos 2θ (v) sin 5θ sin 4θ.
Solution:
1 1
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(i) sin 35 cos 28 ◦ = {sin(35 + 28) + sin(35 − 28) } = {sin(63) + sin(7) }
2 2
1 1
(ii) sin 4x cos 2x = {sin(4x + 2x) + sin(4x − 2x)} = {sin 6x + sin 2x}
2 2
(iii) 2 sin 10θ cos 2θ = sin(10θ + 2θ) + sin(10θ − 2θ) = sin 12θ + sin 8θ
1 1
(iv) cos 5θ cos 2θ = {cos(5θ + 2θ) + cos(5θ − 2θ)} = {cos 7θ + cos 3θ}
2 2
1 1
(v) sin 5θ sin 4θ = {cos(5θ − 4θ) + cos(5θ + 4θ)} = − {cos 9θ − cos θ}
2 2
2. Express each of the following as a product
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◦
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(i) sin 75 − sin 35 ◦ (ii) cos 65 + cos 15 (iii) sin 50 + sin 40 ◦ (iv) cos 35 − cos 75 .
Solution:

75 − 35 75 + 35
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◦
(i) sin 75 − sin 35 ◦ = 2 sin cos = 2 sin 20 cos 55 ◦
2 2

65 + 15 65 − 15
◦
◦
◦
(ii) cos 65 + cos 15 = 2 cos cos = 2 cos 40 cos 25 ◦
2 2

50 + 40 50 − 40
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◦
(iii) sin 50 + sin 40 ◦ = 2 sin cos = 2 sin 45 cos 5 ◦
2 2

75 + 35 75 − 35
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◦
(iv) cos 35 − cos 75 = 2 sin sin = 2 sin 55 sin 20 ◦
2 2