SOLUTION: HI I NEED HELP WITH Sin(X)-0.1Cos(X) = 0.567 Find the angle X Thanks M.samari

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Question 261382: HI I NEED HELP WITH
Sin(X)-0.1Cos(X) = 0.567
Find the angle X
Thanks M.samari
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Sin(X)-0.1Cos(X) = 0.567
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sin(x) - 0.1*(1-sin^2)^(1/2) = 0.567
10sin(x) - (1-sin^2)^(1/2) = 5.67
10sin - 5.67 = (1-sin^2)^(1/2)
100sin^2 - 113.4sin + 32.1489 = 1 - sin^2
101sin^2 - 113.4 sin - 31.1489 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 101x%5E2%2B-113.4x%2B-31.1489+=+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=%28-113.4%29%5E2-4%2A101%2A-31.1489=25443.7156.

Discriminant d=25443.7156 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--113.4%2B-sqrt%28+25443.7156+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-113.4%29%2Bsqrt%28+25443.7156+%29%29%2F2%5C101+=+1.35104387910833
x%5B2%5D+=+%28-%28-113.4%29-sqrt%28+25443.7156+%29%29%2F2%5C101+=+-0.228271601880608

Quadratic expression 101x%5E2%2B-113.4x%2B-31.1489 can be factored:
101x%5E2%2B-113.4x%2B-31.1489+=+%28x-1.35104387910833%29%2A%28x--0.228271601880608%29
Again, the answer is: 1.35104387910833, -0.228271601880608. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+101%2Ax%5E2%2B-113.4%2Ax%2B-31.1489+%29

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sin(x) = -0.228271602
x =~ 193.2 degs, 346.8 degs