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=\frac\pi2,\frac{3\pi}2`

or

`2\sin x+1=0\implies\sin x=-\frac12\implies x=\frac{7\pi}6,\frac{11\pi}6`