# Solve the equation sinx = sin5x

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To solve this equation, we'll move the term from the right side, to the left side:

sinx - sin5x = 0

As we can see, it is a subtraction of 2 trigonometric function of the same kind and it is transforming into a product.

2 cos [(x+5x)/2]*sin [(x-5x)/2]=0

2 cos 3x* sin (-2x)=0

From this product of 2 factors, one or the other factor is zero.

cos 3x=0

This is an elementary equation:

3x = +/-arccos 0 + 2*k*pi

3x=+/- (pi/2)+ 2*k*pi

x=+/- (pi/6)+ 2*k*pi/3, where k is an integer number.

We'll solve the second elementary equation:

sin (-2x)=0

-sin (2x)=0

sin 2x=0

2x=(-1)^k arcsin 0 + k*pi

2x=k*pi

x=k*pi/2

The set of solutions:

**S={+/- (pi/6)+ 2*k*pi/3}or{k*pi/2}**

sinx= sin5x. Or

sin5x-six = 0.

2cos[(5x-x)/2]*[sin[(5x+x)/2] = 0. Or

cos 2x * sin 3x = 0. Or

cos 2x= 0 or sin 3x = 0.

From the cos2x = 0, we get: 2x = 2npi+(-1)^npi/2 or x = npi+(-1)^npi/4.

Fro sin3x=0,

3x = (-1)^n *npi Or x= (-)^npi/3