SOLUTION: Solve for x in the equation cos x - cos 2x = 1/2 where x e (0 degree, 180 degree)

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Question 1199427: Solve for x in the equation cos x - cos 2x = 1/2 where x e (0 degree, 180 degree)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve for x in the equation cos(x) - cos(2x) = 1/2
cos(2x) = 2cos^2(x) - 1
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cos(x) - 2cos^2(x) + 1/2 = 0
2cos^2(x) - cos(x) - 1/2 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-1x%2B-0.5+=+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=%28-1%29%5E2-4%2A2%2A-0.5=5.

Discriminant d=5 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+5+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+5+%29%29%2F2%5C2+=+0.809016994374947
x%5B2%5D+=+%28-%28-1%29-sqrt%28+5+%29%29%2F2%5C2+=+-0.309016994374947

Quadratic expression 2x%5E2%2B-1x%2B-0.5 can be factored:
2x%5E2%2B-1x%2B-0.5+=+%28x-0.809016994374947%29%2A%28x--0.309016994374947%29
Again, the answer is: 0.809016994374947, -0.309016994374947. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-1%2Ax%2B-0.5+%29

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cos(x) = 0.809016994374947
x = 36 degs
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cos(x) = -0.309016994374947
x = 108 degs