Question

The complete general solution of the equation sin6x + sin4x + sin2x = 0 is (nZ)

Answer 1

Given:

The equation sin6x + sin4x + sin2x = 0

To find:

The complete general solution of the equation sin6x + sin4x + sin2x = 0 is?

Solution:

From given, we have,

sin6x + sin4x + sin2x = 0

⇒ (sin6x + sin2x) + sin4x = 0

⇒ 2 sin 4x cos 2x + sin4x = 0

⇒ sin 4x (2cos 2x + 1) = 0

⇒ sin 4x = 0,  2cos 2x + 1 = 0

Now consider, sin 4x = 0

using (sin x = 0 ⇒ x = nπ) we get,

4x = nπ

The general solution is x = nπ/4

Now consider,  2cos 2x + 1 = 0

using (cos x = cos α ⇒ x = 2mx ± α) we get,

cos 2x = cos (2π/3)

2x = 2mπ ± 2π/3

The general solution is x = mπ ± π/3