1. ## solving theta help

$
12sin^2(2\theta)-5sin(2\theta)-3=0
$

My teacher said that there were going to be 8 possible solutions. I don't really understand what steps to take to get 8?

i factored the equation down to

$3sin(2\theta)+1$ and $4sin(2\theta)-3$

i get down to

$sin(2\theta)=-1/3$ and $sin(2\theta)=3/4$

i don't know what to do from here to come out with 8 solutions

also the bounds for the equation are 0 (less than or equal to) theta (less than) 360

2. Hello ldamnedl

Welcome to Math Help Forum!
Originally Posted by ldamnedl
$
12sin^2(2\theta)-5sin(2\theta)-3=0
$

My teacher said that there were going to be 8 possible solutions. I don't really understand what steps to take to get 8?

i factored the equation down to

$3sin(2\theta)+1$ and $4sin(2\theta)-3$

i get down to

$sin(2\theta)=-1/3$ and $sin(2\theta)=3/4$

i don't know what to do from here to come out with 8 solutions

also the bounds for the equation are 0 (less than or equal to) theta (less than) 360
Your working is fine. Note next, that if $0^o\le\theta<360^o$, then $0^o\le2\theta<720^o$. So we have two equations to solve for values of $2\theta$ in this range; then divide these answers by $2$ to get the values of $\theta$. I'll start you off:
$\sin2\theta = -\frac13$

$\Rightarrow 2\theta = 180 + 19.47,\; 360 - 19.47,\; 540+19.47,\; 720 - 19.47$
$= 199.47, ...$
$\Rightarrow \theta = 99.7^o, ...$
Can you complete it now?