what are all values of theta in sin 2 theta = sin theta

Question

find all values of θ in the interval 0°≤θ<360, that satisfies the equation sin(2θ)= sin θ

Answer 1

sin 2θ = sin θ

But we know a formula for sin 2θ = 2 sin θ cosθ

Therefore we must solve 2 sin θ cosθ = sin θ.

 

Rearrange & factor:  (sin θ) (2 cos θ - 1) = 0.

 

Using the Zero Product Property, your solutions are those which satisfy either of the following:

(sin θ) = 0   OR   (2 cos θ - 1)= 0.

 

You can take it from here!

Answer 2

sin 2θ = sin θ

2 sin θ cos θ = sin θ          double angle formula

2 sin θ cos θ - sin θ = 0     subtract sin θ from both sides

sin θ (2 cos θ - 1) = 0        GCF

 

sin θ = 0                            zero product rule

 

2 cos θ - 1 = 0                  zero product rule

2 cos θ = 1

cos θ = 0.5

θ = π/3                    

θ = 5π/3                 

Answer 3

I am unaware of any general rule, so we are forced to resort to checking and guessing. We can cleverly pick out simple angles between 0 and 360 and check whether or not the sine of those angles equals the sine of their doubles.

 

It turns out that 

 

θ = 0, 60, and 180

 

all work. 

 

sin[2*(0)] = sin(0) = 0

 

sin[2*(60)] = sin(60) = 0.866

 

sin[2*(180)] = sin(180) = 0

 

There may be more. Keep testing angles.