SOLUTION: Find all solutions of the equation in the interval [0,2π). {{{ 4cos^2(x)=5 - 4 sin(x) }}} Write your answer in radians in terms of π. If there is more than one s

Algebra ->  Trigonometry-basics -> SOLUTION: Find all solutions of the equation in the interval [0,2π). {{{ 4cos^2(x)=5 - 4 sin(x) }}} Write your answer in radians in terms of π. If there is more than one s      Log On


   

Question 985594: Find all solutions of the equation in the interval [0,2π).

+4cos%5E2%28x%29=5+-+4+sin%28x%29+
Write your answer in radians in terms of π.
If there is more than one solution, separate them with commas.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find all solutions of the equation in the interval [0,2π).

+4cos%5E2%28x%29=5+-+4+sin%28x%29+
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Sub 1 - sin^2 for cos^2, then it's a quadratic.
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4 - 4sin^2 = 5 - 4sin
4sin^2 - 4sin + 1 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B-4x%2B1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A4%2A1=0.

Discriminant d=0 is zero! That means that there is only one solution: x+=+%28-%28-4%29%29%2F2%5C4.
Expression can be factored: 4x%5E2%2B-4x%2B1+=+%28x-0.5%29%2A%28x-0.5%29

Again, the answer is: 0.5, 0.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B-4%2Ax%2B1+%29

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sin(x) = 1/2
x = pi/6, 5pi/6