HI everyone, I'm not sure how to start this question and I need some help!
B) Solve this equation algebraically where 0<x<2pi. Give answers as exact solutions.
Sqrt(2)Cos^2x - cosx =0
Can I factor a Cosx out? Or does the Root 2 prevent me from doing that? I'm so confused! Thanks for any help.
MathBioMajor
Factoring out cos x will make things much more complicated. When working problems like this, the most direct route is usually the best.
√(2cos x) - cos x = 0.
Move the second term over to the right side:
√(2 cos² x) = cos x.
Now square both sides of the last equation:
2 cos² x = cos² x.
Now move the right term back over to the left side, and combine like terms:
2 cos² x - cos² x = 0
cos² x = 0.
Solve for x by taking the square root of both sides of the last equation:
√(cos² x) = √0
cos x = 0.
There are two angles in the given interval with cosine = 0. They are π/2 (90°), and 3π/2 (270°).
So the answers are x = π/2, and x = 3π/2.