how to do 2sin^2theta=3(1-costheta)

Question

I have a problem with identies and the way teacher asks us to do stuff we never went over in class help

Answer 1

Test the identity:

Let θ=30°, then sinθ=½ and cosθ=√3/2, so 2sin²θ=2/4=½; and 3(1-cosθ)=(3/2)(2-√3).

So the identity is not true. Let's see what is true:

2sin²θ=2(1-cos²θ)=2(1-cosθ)(1+cosθ).

3(1-cosθ)=3(1-(1-2sin²(½θ))=6sin²(½θ), so:

6sin²(½θ)=3(1-cosθ), sin²(½θ)=½(1-cosθ).

If 2sin²θ=3(1-cosθ) is an equation instead of an identity it can be solved for θ.

2sin²θ=2(1-cos²θ)=3(1-cosθ),

2-2cos²θ=3-3cosθ,

2cos²θ-3cosθ+1=(2cosθ-1)(cosθ-1), so cosθ=½ or 1.

If θ is between 0° and 360°, the solution is θ=0°, 60° or 300°.