SOLUTION: solve 3 sin^2 theta=4cos theta-1

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Question 1046316: solve 3 sin^2 theta=4cos theta-1
Found 2 solutions by t0hierry, ikleyn:
Answer by t0hierry(194) About Me  (Show Source):
You can put this solution on YOUR website!
3sin^2 theta = 3 - 3 cos^2 theta = 4 cos theta - 1
or
cos^2 theta + 4/3 cos theta - 4/3 = 0
cos theta = 1/2[-4/3 +- sqrt(16/9 + 16*3/3*3)]
cos theta = -4/6 + 1/2 4/3 sqrt 5
cos theta = 2/3(sqrt(5) - 1)
Can you check?

Answer by ikleyn(52909) About Me  (Show Source):
You can put this solution on YOUR website!
.
solve 3 sin^2 theta=4cos theta-1
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

3%2Asin%5E2%28theta%29 = 4%2Acos%28theta%29-1

Replace sin%5E2%28theta%29 by 1-cos%5E2%28theta%29 to get the equation uniform for cos%28theta%29:

3%281-cos%5E2%28theta%29%29 = 4%2Acos%28theta%29-1,

3-3%2Acos%5E2%28theta%29 = 4%2Acos%28theta%29-1,

3cos%5E2%28theta%29+%2B+4cos%28theta%29+-+4 = 0.


Factor left side:


(3cos(x)-2)*(cos(x)+2) = 0.


The equation deploys in two independent equations


1.  3cos%28theta%29+-+2 = 0  --->  cos%28theta%29 = 2%2F3  --->  theta = +/-arccos%282%2F3%29+%2B+2k%2Api,  where k is any integer,  and


2.  cos%28theta%29+%2B+2 = 0  --->  cos%28theta%29 = -2,  which has no solutions.


Answer.  The solutions are  theta = +/-arccos%282%2F3%29+%2B+2k%2Api,  where k is any integer.




Plot y = 3%2Asin%5E2%28x%29-4cos%28x%29%2B1