How do you solve sin 2theta sin theta = cos theta?

1 Answer
May 16, 2015

From the formula for sin (2theta) we have

sin 2theta = 2sin theta cos theta

From Pythagoras theorem we have

sin^2 theta + cos^2 theta = 1

So

sin 2theta sin theta
= (2sin theta cos theta)sin theta
= 2sin^2 theta cos theta
=2(1-cos^2 theta)cos theta

Putting this together with your equation, we get

2(1-cos^2 theta)cos theta = cos theta

If cos theta = 0 then both sides will be zero.
So some solutions to the original problem are:

theta = pi/2 + npi for all n in ZZ.

On the other hand, if cos theta != 0, divide both sides of the equation by cos theta to get

2(1-cos^2 theta) = 1

Divide both sides by 2 to get

1-cos^2 theta = 1/2

So cos^2 theta = 1/2 and cos theta = +-1/sqrt(2)

This is true for

theta = pi/4 + (npi)/2 for all n in ZZ.