SOLUTION: Exact values. 2-2 cos^2x=-3sin x-1

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Question 1008640: Exact values. 2-2 cos^2x=-3sin x-1
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
2-2cos%5E2%28x%29=-3sin%28x%29-1

Add 1 to both sides:

3-2cos%5E2%28x%29=-3sin%28x%29

Use the identity sin%5E2%28theta%29%2Bcos%5E2%28theta%29=1 solved for cos2(q)

3-2%281-sin%5E2%28x%29%5E%22%22%29=-3sin%28x%29

3-2%2B2sin%5E2%28x%29=-3sin%28x%29

1%2B2sin%5E2%28x%29=-3sin%28x%29

Get 0 on the right side

2sin%5E2%28x%29%2B3sin%28x%29%2B1=0

Factor the left side as a quadratic in sin(x)

%282sin%28x%29%5E%22%22%2B1%5E%22%22%29%28sin%28x%29%5E%22%22%2B1%5E%22%22%29=0

Use the zero-factor property:

2sin%28x%29%2B1=0;   sin%28x%29%2B1=0

2sin%28x%29=-1;    sin%28x%29=-1

sin%28x%29=-1%2F2;     x=3pi%2F2

x=7pi%2F6;11pi%2F6   

These are the solutions for 0%3C=x%3C2pi.
To get all solutions add 2pi%2An to them.

Edwin