SOLUTION: find all values of feta, to the nearest degree, that satisfy the equation 7cos cubed feta=5cos squared feta+cos feta in the interval 0 degrees to 360 degrees

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Question 723880: find all values of feta, to the nearest degree, that satisfy the equation 7cos cubed feta=5cos squared feta+cos feta in the interval 0 degrees to 360 degrees
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
7cos%5E3%28theta%29=5cos%5E2%28theta%29%2Bcos%28theta%29 --> 7cos%5E3%28theta%29-5cos%5E2%28theta%29-cos%28theta%29=0
So that I do not have to keep writing cos%28theta%29 so many times, I will define a shorter-named variable
x=cos%28theta%29
Using the newly defined variable I can write the equation as
7x%5E3-5x%5E2-x=0
That is much easier to type.
Now, we solve for x:
7x%5E3-5x%5E2-x=0 --> x%287x%5E2-5x-1%29=0 (taking out x as common factor)
The solutions are x=0 and the solutions to the quadratic equation
7x%5E2-5x-1=0 , which we can find by using the quadratic formula:
x+=+%28-%28-5%29+%2B-+sqrt%28%28-5%29%5E2-4%2A7%2A%28-1%29+%29%29%2F%282%2A7%29+
x+=+%285+%2B-+sqrt%2825%2B28%29%29%2F14
x+=+%285+%2B-+sqrt%2853%29%29%2F14
The two values we get from the quadratic formula are approximately
x=0.877 and x=-0.163 (rounding)
x=0 corresponds to highlight+%28theta=90%5Eo%29 and highlight+%28theta=270%5Eo%29 for theta in the interval between 0%5Eo and 360%5Eo , because those are the only two angles with zero cosine in that interval.
From cos%28theta%29=%285+%2B+sqrt%2853%29%29%2F14 we get highlight+%28theta=28.7%5Eo%29 (rounded) and
theta=360%5Eo-28.7%5Eo=highlight+%28331.3%5Eo%29 (also a rounded approximation).
From cos%28theta%29=%285+-+sqrt%2853%29%29%2F14 we get highlight+%28theta=99.4%5Eo%29 (rounded) and
theta=360%5Eo-99.4%5Eo=highlight+%28260.6%5Eo%29 (also a rounded approximation).