SL 3.8—Trigonometric equations

Solve trigonometric equations on a stated interval.

Find a reference angle, use quadrant symmetry, and list only solutions inside the requested interval.

Worked example

sin x=1/2 on [0,2π] gives x=π/6 and 5π/6.

The interval determines which repeated solutions are included.

A calculator principal value is not the complete interval solution.

Quadratic reduction example on $0\le x\le2\pi$: $2\sin^2x-3\sin x+1=0$ factors as $(2\sin x-1)(\sin x-1)=0$. Hence $\sin x=1/2$ or $\sin x=1$, giving $x=\pi/6,\pi/2,5\pi/6$. Check every candidate in the original equation and report only values in the stated finite interval; no general solution is required.