Question

Solve for x where... os(2x) - sin(x) = 0" alt="cos(2x) - sin(x) = 0" align="absmiddle" class="latex-formula">

Answer 1

Use double angle identity:
$cos (2x) = 1-2sin^2 (x)$
After substituting and moving to Right side of equation:
$2 sin^2 (x) + sin (x) -1 = 0$
This is a quadratic, which can be solved by factoring or quadratic formula.
It may be helpful to replace sin(x) with another variable, then solve.
$u = sin(x) \\ 2u^2 +u-1=0 \\ (2u-1)(u+1) = 0 \\ u = \frac{1}{2} , -1$
Now substitute sin(x) back in, we have
$sin (x) = \frac{1}{2}, -1$
Using a unit circle or inverse sine function in calculator we can find the appropriate angles.
$x = \frac{\pi}{6}, \frac{3\pi}{2}$