Answer:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}\frac{2\pi}{3}\frac{4\pi}{3}[/tex]
Step-by-step explanation:
You need 2 things in order to solve this equation: a trig identity sheet and a unit circle.
You will find when you look on your trig identity sheet that
[tex]cos(2\theta)=1-2sin^2(\theta)[/tex]
so we will make that replacement, getting everything in terms of sin:
[tex]sin(\theta)+1=1-2sin^2(\theta)[/tex]
Now we will get everything on one side of the equals sign, set it equal to 0, and solve it:
[tex]2sin^2(\theta)+sin(\theta)=0[/tex]
We can factor out the sin(theta), since it's common in both terms:
[tex]sin(\theta)(2sin(\theta)+1)=0[/tex]
Because of the Zero Product Property, either
[tex]sin(\theta)=0[/tex] or
[tex]2sin(\theta)+1=0[/tex]
Look at the unit circle and find which values of theta have a sin ratio of 0 in the interval from 0 to 2pi. They are:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2}[/tex]
The next equation needs to first be solved for sin(theta):
[tex]2sin(\theta)+1=0[/tex] so
[tex]2sin(\theta)=-1[/tex] and
[tex]sin(\theta)=-\frac{1}{2}[/tex]
Go back to your unit circle and find the values of theta where the sin is -1/2 in the interval. They are:
[tex]\theta=\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
