Pre-calc 12 problem? Need help?

Question

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 · Answer

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.

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