Question

For sin2x+cosx=0, use a double-angle or half-angle formula to simplify the equation and then find all solutions of the equation in the interval [0,2π).

Answer 1

The double-angle formula for sine gives

$\sin(2x)=2\sin x\cos x$

so

$\sin(2x)+\cos x=2\sin x\cos x+\cos x=\cos x(2\sin x+1)=0$

Then either

$\cos x=0\implies x=\dfrac\pi2,\dfrac{3\pi}2$

or

$2\sin x+1=0\implies\sin x=-\dfrac12\implies x=\dfrac{7\pi}6,\dfrac{11\pi}6$