SOLUTION: solve the equation for solutions in the interval from [0,360 degrees): sin(2 theta) +1=cos(2 theta)

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Question 918939: solve the equation for solutions in the interval from [0,360 degrees): sin(2 theta) +1=cos(2 theta)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
solve the equation for solutions in the interval from [0,360 degrees): sin(2 theta) +1=cos(2 theta)
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sin + 1 = cos = 1 - sin^2
sin^2 + sin - 1 = 0
x = sin(2theta)
x^2 + x - 1 = 0
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Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B1x%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=%281%29%5E2-4%2A1%2A-1=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-%281%29%2Bsqrt%28+5+%29%29%2F2%5C1+=+0.618033988749895
x%5B2%5D+=+%28-%281%29-sqrt%28+5+%29%29%2F2%5C1+=+-1.61803398874989

Quadratic expression 1x%5E2%2B1x%2B-1 can be factored:
1x%5E2%2B1x%2B-1+=+%28x-0.618033988749895%29%2A%28x--1.61803398874989%29
Again, the answer is: 0.618033988749895, -1.61803398874989. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-1+%29

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x =~ 0.618034 = 2theta
2theta =~ 38.17, 321.83, 398.18, 681.83 degs
theta = 19.085, 160.91, 199.09, 340.91 degs