Trig Equations

Multiple-Angle Trig Equation

Solve sin(2x) = c on [0, 2π) by substituting u = 2x, finding all u ∈ [0, 4π), then halving each solution to get x.

multiple-angle Substitution: let u = 2x. Then sin(u) = c on u ∈ [0, 4π). Find all u solutions (one period gives 2; two periods give 4). Divide each u by 2 to get x ∈ [0, 2π).

Example

Solve for the multiple angle first, then divide to get all solutions.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

sin⁡(2x)=12x∈[0,2π)\sin( 2 x)= \frac{1}{2} \quad x\in[ 0 , 2 \pi )

Step 2 — Solve for the angle

Let u equal 2x, so the angle interval doubles to 4 pi.

u=2xu∈[0,4π)u= 2 x\quad u\in[ 0 , \hl{4} \pi )

Step 3 — Angle solutions

Find all 4 angle solutions for u.

u∈{π6,5π6,13π6,17π6}u\in\{ \hlmath{\frac{\pi}{6}} , \hlmath{\frac{5\pi}{6}} , \hlmath{\frac{13\pi}{6}} , \hlmath{\frac{17\pi}{6}} \}

Step 4 — Divide by 2

Divide each angle by 2 to get all 4 x-solutions.

x∈{π12,5π12,13π12,17π12}x\in\{ \hlmath{\frac{\pi}{12}} , \hlmath{\frac{5\pi}{12}} , \hlmath{\frac{13\pi}{12}} , \hlmath{\frac{17\pi}{12}} \}