SOLUTION: If sin xcox+sin2x=1, what are the values of x?

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Question 770315: If sin xcox+sin2x=1, what are the values of x?
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
If sin xcox+sin2x=1, what are the values of x?
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sin(x)cos(x) = sin(2x)/2
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sin^2(2x) + sin(2x)/2 - 1 = 0
Sub u for sin(2x)
u%5E2+%2B+u%2F2+-+1+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B0.5x%2B-1+=+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=%280.5%29%5E2-4%2A1%2A-1=4.25.

Discriminant d=4.25 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-0.5%2B-sqrt%28+4.25+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%280.5%29%2Bsqrt%28+4.25+%29%29%2F2%5C1+=+0.780776406404415
x%5B2%5D+=+%28-%280.5%29-sqrt%28+4.25+%29%29%2F2%5C1+=+-1.28077640640442

Quadratic expression 1x%5E2%2B0.5x%2B-1 can be factored:
1x%5E2%2B0.5x%2B-1+=+%28x-0.780776406404415%29%2A%28x--1.28077640640442%29
Again, the answer is: 0.780776406404415, -1.28077640640442. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B0.5%2Ax%2B-1+%29

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Ignore the >1 answer
u =~ 0.780776
sin(2x) = 0.780776
2x = 0.8959,2.2457 + 2n*pi, n = 0,1,2,3...
x = 0.4482,1.1228 + n*pi, n = 0,1,2,3...