What is x if sin8x=sin5x ?

We'll recall the method of solving the elementary
trigonometric equation:

sin x = a => x =
(-1)^k*arcsin a + k*pi

Comparing, we'll
get:

8x = (-1)^k*arcsin (sin 5x) +
k*pi

But arcsin (sin 5x) =
5x

8x = (-1)^k* 5x +
k*pi

We'll discuss two
cases:

1) If k is an even integer
number:

k = 2n

8x = (-1)^2n*
5x + 2n*pi

8x = 5x + 2n*pi

8x
- 5x = 2n*pi

3x = 2n*pi

x =
2n*pi/3

2) If k is an odd integer
number:

k = 2n + 1

8x =
(-1)^(2n+1)* 5x + (2n+1)*pi

8x = -5x + 2n*pi +
pi

8x + 5x = 2n*pi + pi

13x =
2n*pi + pi

x = 2n*pi/13 +
pi/13

The solutions of the equation are
represented by the reunion of the sets: {2n*pi/3/ n is an integer number} U
{(2n+1)*pi/13 / n is an integer number}.