Page 52 - mathsvol1ch1to3ans

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3. Show that sin 12 sin 48 sin 54 = .
8
Solution:

1
sin 12 (sin 48 cos 36) = (cos 36 − cos 60) cos 36
2

1 1
2
= cos 36 − cos 36
2 4
√ √
   
! 2
1  5 + 1 1 5 + 1 1
=   −  =
4 4 4 8
2 
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π 2π 3π 4π 5π 6π 7π 1
4. Show that cos cos cos cos cos cos cos = .
15 15 15 15 15 15 15 128
Solution:
π
cos π cos 2π cos 3π cos 4π cos 5π cos 6π cos 7π = cos π cos 2π cos 3π cos 4π cos cos 6π cos 7π
15 15 15 15 15 15 15 15 15 15 15 3 15 15
= cos π cos 2π cos 3π cos 4π 1 cos 6π cos π − 8π
15 15 15 15 2 15 15
1
= − cos π cos 2π cos 4π cos 8π cos 3π cos 6π
2 15 15 15 15 15 15
= − 1 sin 2π sin 4π sin 8π sin 16π 3π cos 6π
15
15
8π cos
15
15
2 2 sin π 2 sin 2π 2 sin 4π 2 sin 15 15
15 15 15 15
= − 1 sin 16π π cos 3π cos 2 3π
15
4
2 2 sin 15 15
15
sin(π+ ) sin 6π sin 12π
π
1 15 15 15
= − 5 π 3π 6π
2 sin 2 sin 2 sin
15 15 15
sin(π− )
3π
= 1 15
2 7 sin 3π
15
= 1
128
sin 8x cos x − sin 6x cos 3x
5. Show that = tan 2x.
cos 2x cos x − sin 3x sin 4x
Solution:
sin 8x cos x − sin 6x cos 3x 2 sin 8x cos x − 2 sin 6x cos 3x
=
cos 2x cos x − sin 3x sin 4x 2 cos 2x cos x − 2 sin 3x sin 4x
sin 9x + sin 7x − sin 9x − sin 3x
=
cos 3x + cos x − cos x + cos 7x
sin 7x − sin 3x
=
cos 3x + cos 7x
2 sin 2x cos 5x
=
2 cos 5x cos 2x
= tan 2x
(cos θ − cos 3θ) (sin 8θ + sin 2θ)
6. Show that = 1.
(sin 5θ − sin θ) (cos 4θ − cos 6θ)
Solution:
(cos θ − cos 3θ) (sin 8θ + sin 2θ) (2 sin 2θ sin θ) (2 sin 5θ cos 3θ)
= = 1
(sin 5θ − sin θ) (cos 4θ − cos 6θ) (2 sin 2θ cos 3θ) (2 sin 5θ sin θ)
7. Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x).
Solution: