SOLUTION: Solve for x: 4sin^2x+sinx+3=6cos^2x

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Question 1081158: Solve for x: 4sin^2x+sinx+3=6cos^2x
Answer by ikleyn(52908) About Me  (Show Source):
You can put this solution on YOUR website!
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4sin^2x+sinx+3=6cos^2x
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4%2Asin%5E2%28x%29+%2B+sin%28x%29+%2B+3 = 6%2Acos%5E2%28x%29  ---> (replace cos^2(x) by 1-sin^2(x))  --->

4%2Asin%5E2%28x%29+%2B+sin%28x%29+%2B+3 = 6%2A%281-sin%5E2%28x%29%29,

4%2Asin%5E2%28x%29+%2B+sin%28x%29+%2B+3 = 6-6%2Asin%5E2%28x%29,

10%2Asin%5E2%28x%29+%2B+sin%28x%29+-+3 = 0.

Introduce new variable y = sin(x). Then the last equation takes the form

10y%5E2+%2B+y+-+3 = 0.

Its roots, according to the quadratic formula, are

y%5B1%2C2%5D = %28-1+%2B-+sqrt%28+1%2B4%2A10%2A3%29%29%2F20 = %28-1+%2B-+11%29%2F20.


1.  y%5B1%5D = %28-1+%2B+11%29%2F20 = 1%2F2  --->  sin(x) = 1%2F2  --->  x = pi%2F6  and/or  x = 5pi%2F6.


2.  y%5B2%5D = %28-1+-+11%29%2F20 = -3%2F5  --->  sin(x) = -3%2F5  --->  x = -arcsin%283%2F5%29  and/or  x = pi%2Barcsin%283%2F5%29.


Answer.  The original equation has 4 solutions:  pi%2F6, 5pi%2F6, -arcsin%283%2F5%29 and pi%2Barcsin%283%2F5%29.




Plot y = 4%2Asin%5E2%28x%29%2Bsin%28x%29%2B3 (red) and y = 6%2Acos%5E2%28x%29 (green)