Find the general solution for each of the following equations: sin x + sin 3x + sin 5x = 0

Solution:

sin x + sin 3x + sin 5x = 0

(sin 5x + sin x) + sin 3x = 0

We have  sin A + sin B = 2sin {(A + B) / 2} cos {(A - B) / 2}. So we get

2 sin {(5x + x) / 2} cos {(5x - x) / 2} + sin 3x = 0

2 sin 3x cos 2x + sin 3x = 0

sin 3x(2cos 2x + 1) = 0

Now, sin 3x = 0 or 2cos 2x + 1 = 0

sin 3x = 0 or cos 2x = -1/2

sin 3x = sin 0 or cos 2x = 2π/3

3x = nπ or 2x = (2nπ ± 2π/3), where n∈Z.

x = nπ/3 or x = (nπ ± π/3), where n∈Z.

NCERT Solutions Class 11 Maths Chapter 3 Exercise 3.4 Question 9

Find the general solution for each of the following equations: sin x + sin 3x + sin 5x = 0

Summary:

The general solutions of the equation sin x + sin 3x + sin 5x = 0 are x = nπ/3 or (nπ ± π/3), where n∈Z.