Question

find solutions between 0 & 2pi : cos(x/2)-sin(x)=0

Answer 1

use half angle identity
\[\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}\]
then move sin x to right side , square both sides
\[\frac{1+ \cos x}{2} = \sin^2 x\]
substitute using pythagorean identity
\[\frac{1+\cos x}{2} = 1 - \cos^2 x\]
solve the quadratic for cos x
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Answer 2

Transform the equation into a product of 2 basic trig equation.
cos x/2 - sin x = 0. Replace sin x = 2cos x/2*sin x/2 (trig identity sin 2a)
cos x/2 (1 - 2sin x/2) = 0. Solve the 2 basic trig equations.
1) cos x/2 = 0 --> x/2 = 0, and Pi --> x = 0, and 2Pi
2) sin x/2 = 1/2 --> x/2 = Pi/6, and 5Pi/6 --> x = Pi/3 and 5Pi/3
Answers between (0, 2Pi): 0, Pi/3; 5Pi/3; 2Pi