SOLUTION: Find all solutions in the interval [0,2pi) Cos^2 theta -9 cos theta - 1 =0 (Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers

Algebra ->  Trigonometry-basics -> SOLUTION: Find all solutions in the interval [0,2pi) Cos^2 theta -9 cos theta - 1 =0 (Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers       Log On


   

Question 1197838: Find all solutions in the interval [0,2pi)
Cos^2 theta -9 cos theta - 1 =0
(Type your answer in radians. Round to four decimal places as needed. Use a comma to separate answers as needed.)
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find all solutions in the interval [0,2pi)
Cos^2 theta -9 cos theta - 1 =0
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If you mean Cos^2(t) -9cos(t) - 1 = 0
Sub x for cos(t)
x%5E2+-+9x+-+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%2B-9x%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=%28-9%29%5E2-4%2A1%2A-1=85.

Discriminant d=85 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--9%2B-sqrt%28+85+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-9%29%2Bsqrt%28+85+%29%29%2F2%5C1+=+9.10977222864644
x%5B2%5D+=+%28-%28-9%29-sqrt%28+85+%29%29%2F2%5C1+=+-0.109772228646444

Quadratic expression 1x%5E2%2B-9x%2B-1 can be factored:
1x%5E2%2B-9x%2B-1+=+%28x-9.10977222864644%29%2A%28x--0.109772228646444%29
Again, the answer is: 9.10977222864644, -0.109772228646444. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-9%2Ax%2B-1+%29

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Ignore the value >1, no real solution.
cos(t) =~ -0.109772228
theta = ~96.3022 degs
theta = ~263.6978 dega
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Convert to radians if you like.