SOLUTION: Find all solutions in the interval [0,360) cos^2(theta)+8cos(theta)+5=0

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Question 1088061: Find all solutions in the interval [0,360)
cos^2(theta)+8cos(theta)+5=0
Answer by ikleyn(52800) About Me  (Show Source):
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Find all solutions in the interval [0,360)
cos^2(theta)+8cos(theta)+5=0
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cos%5E2%28theta%29+%2B+8cos%28theta%29+%2B+5 = 0.

Introduce new variable x = cos%28theta%29. Then your equation takes the form

x%5E2+%2B+8x+%2B+5 = 0.


Apply the quadratic formula to solve for x:

x%5B1%2C2%5D = %28-8+%2B-+sqrt%288%5E2+-+4%2A1%2A5%29%29%2F2 = %28-8+%2B-+sqrt%2844%29%29%2F2 = -4+%2B-+sqrt%2811%29.


1)  x%5B1%5D = -4+%2B+sqrt%2811%29%29 =~ -0.68.

    cos%28theta%29 = -4+%2B+sqrt%2811%29  ====>  theta = arccos%28-4+%2B+sqrt%2811%29%29  and  theta = 2pi+-+arccos%28-4+%2B+sqrt%2811%29%29.



2)  x%5B2%5D = -4+-+sqrt%2811%29%29 < -1.

    There is NO theta satisfying cos%28theta%29 = x%5B2%5D.


Answer. The original equation has two solutions  theta = arccos%28-4+%2B+sqrt%2811%29%29  and  theta = 2pi+-+arccos%28-4+%2B+sqrt%2811%29%29.

Solved.


If you want to see more solved problems/samples on trigonometric equations, look into the lessons
    - Solving simple problems on trigonometric equations
    - Solving typical problems on trigonometric equations
    - Solving more complicated problems on trigonometric equations
    - Solving advanced problems on trigonometric equations
    - OVERVIEW of lessons on calculating trig functions and solving trig equations
in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Trigonometry: Solved problems".