Answer:
[tex]\theta=\frac{\pi}{2},\frac{3\pi}{2},\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
Step-by-step explanation:
If I'm interpreting that correctly, you are trying to solve this equation:
[tex]sin(\theta )+1=cos(2\theta)[/tex]
for theta. To do this, you will need a trig identity sheet (I'm assuming you got one from class) and a unit circle (ditto on the class thing).
We need to solve for theta. If I look to my trig identities, I will see a double angle one there that says:
[tex]cos(2\theta)=1-2sin^2(\theta)[/tex]
We will make that replacement, then we will have everything in terms of sin.
[tex]sin(\theta)+1=1-2sin^2(\theta)[/tex]
Now get everything on one side of the equals sign to solve for theta:
[tex]2sin^2(\theta)+sin(\theta)=0[/tex]
We can factor out the common sin(theta):
[tex]sin\theta(2sin\theta+1)=0[/tex]
By the Zero Product Property, either
[tex]sin\theta=0[/tex] or
[tex]2sin\theta+1=0[/tex]
Now look at your unit circle and find that the values of theta where the sin is 0 are located at:
[tex]\theta=\frac{\pi }{2},\frac{3\pi}{2}[/tex]
The next one we have to solve for theta:
[tex]2sin\theta+1=0[/tex] simplifies to
[tex]2sin\theta=-1[/tex] and
[tex]sin\theta=-\frac{1}{2}[/tex]
Look at the unit circle again to find the values of theta where the sin is -1/2:
[tex]\theta=\frac{2\pi}{3},\frac{4\pi}{3}[/tex]
Those ar your values of theta!
