SOLUTION: I need to solve the equation cos^2(theta) = 4sin(theta)+4, on the interval 0<(theta)<2pi. I thought I had simplified it correctly using trig identities, but something went wrong.

Algebra ->  Trigonometry-basics -> SOLUTION: I need to solve the equation cos^2(theta) = 4sin(theta)+4, on the interval 0<(theta)<2pi. I thought I had simplified it correctly using trig identities, but something went wrong.       Log On


   

Question 72889: I need to solve the equation cos^2(theta) = 4sin(theta)+4, on the interval 0<(theta)<2pi. I thought I had simplified it correctly using trig identities, but something went wrong. Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
use the identity %28cos%28x%29%29%5E2=1-%28sin%28x%29%29%5E2 and replace %28cos%28x%29%29%5E2
1-%28sin%28theta%29%29%5E2=4sin%28theta%29%2B4
Now let x=sin%28theta%29
1-x%5E2=4x%2B4
Get x to one side
x%5E2%2B4x%2B3=0
Solve for x
%28x%2B3%29%28x%2B1%29=0
Solutions are x=-3 and x=-1
This means sin%28theta%29=-3 and sin%28theta%29=-1
Since the first solution doesn't make any sense, ignore it. Solve for sin%28theta%29=-1
arcsin%28sin%28theta%29%29=arcsin%28-1%29
theta=-pi%2F2%2Bpi%2AnSince -pi%2F2 is not in the interval 0<(theta)<2pi it can be ignored also
Now find the other value
theta=pi-arcsin%28-1%29This allows us to go the other value of sine
theta=pi-%28-pi%2F2%29
theta=%282pi%2Bpi%29%2F2
theta=3pi%2F2There's our other value
So theta=3pi%2F2 is our answer.


Check:
%28cos%283pi%2F2%29%29%5E2+=+4sin%283pi%2F2%29%2B4
0=-4%2B4
0=0Our answer works and is in the interval 0<(theta)<2pi