SOLUTION: Solve cosec 2θ=secθ+cotθ

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Question 624676: Solve cosec 2θ=secθ+cotθ
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve cosec 2θ=secθ+cotθ
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csc(2t) = sec(t) + cot(t)
1/sin(2t) = 1/cos(t) + cos(t)/sin(t)
1/(2sin*cos) = 1/cos + cos/sin
1/2 = sin + cos^2
1/2 = sin + (1 - sin^2)
2sin^2 - 2sin - 1 = 0
2x^2 - 2x - 1 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B2x%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=%282%29%5E2-4%2A2%2A-1=12.

Discriminant d=12 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+12+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%282%29%2Bsqrt%28+12+%29%29%2F2%5C2+=+0.366025403784439
x%5B2%5D+=+%28-%282%29-sqrt%28+12+%29%29%2F2%5C2+=+-1.36602540378444

Quadratic expression 2x%5E2%2B2x%2B-1 can be factored:
2x%5E2%2B2x%2B-1+=+%28x-0.366025403784439%29%2A%28x--1.36602540378444%29
Again, the answer is: 0.366025403784439, -1.36602540378444. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B2%2Ax%2B-1+%29

sin(t) = x =~ 0.36603
theta =~ 21.47 degs (principal value)