Question

Number of solutions of the trigonometric equation in [ 0 , pi] sin 3 theta = 4 sin theta* sin 2 theta* sin 4 theta a) 4 b) 6 c) 8 d)10 PLS ANSWER THIS QUESTION AND GIVE ALL THE STEPS AND PLS DONT LINK TO ANY OTHER QUESTION

Answer 1

sin 3x = 4 sin x  * sin 2x * sin 4x
3 sin x - 4 sin³ x = 4 sin x * sin 2x * sin 4x

either sin x = 0    ie.,  x = 0  or   π
OR,   3 - 4 sin² x = 4  sin 2x  * 2 sin 2x cos 2x
         1 + 2 cos 2x = 8 sin² 2x * cos 2x
         1 = 2 cos 2x (4 sin² 2x - 1)
          1 = (2 Cos 2x) (3 - 4 cos² 2x)
         8 Cos³ 2x - 6 cos 2x + 1 = 0

This is a polynomial in cos 2x of degree 3.  as x varies for 0 to π, cos 2x varies from -1 to 1.   The changes in sign of coefficients imply that there are at most 2 solutions for  cos 2 x > 0.  There is at most 1 solution for cos 2x < 0.   There are 3 solutions for cos 2x...

For each solution of cos 2x, there are two solutions for cos x. as cos 2x is a quadratic in cos x.   For each solution of cos x there is one solution for x in [0, π].

Hence, there are possibly 2 + 2 *3 = 8 solutions in all.
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The solutions are: 0, π/9, 2π/9, 4π/9, 5π/9, 7π/9, 8π/9, π..

Answer 2

sin 3x = 4 sin x * sin 2x * sin 4x

3 sin x - 4 sin³ x = 4 sin x * sin 2x * sin 4x

either sin x = 0 ie., x = 0 or π

OR, 3 - 4 sin² x = 4 sin 2x * 2 sin 2x cos 2x

1 + 2 cos 2x = 8 sin² 2x * cos 2x

1 = 2 cos 2x (4 sin² 2x - 1)

1 = (2 Cos 2x) (3 - 4 cos² 2x)

8 Cos³ 2x - 6 cos 2x + 1 = 0

This is a polynomial in cos 2x of degree 3. as x varies for 0 to π, cos 2x varies from -1 to 1. The changes in sign of coefficients imply that there are at most 2 solutions for cos 2 x > 0. There is at most 1 solution for cos 2x < 0. There are 3 solutions for cos 2x...

For each solution of cos 2x, there are two solutions for cos x. as cos 2x is a quadratic in cos x. For each solution of cos x there is one solution for x in [0, π].

Hence, there are possibly 2 + 2 *3 = 8 solutions in all.

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The solutions are: 0, π/9, 2π/9, 4π/9, 5π/9, 7π/9, 8π/9, π..