Question

In Problems $69-78$, solve each equation on the interval $0\le \theta <2\mathrm{\pi }$.

$\sin \left(2\theta \right)=\cos \theta$.

Short answer

The solution set for the equation$\sin \left(2\theta \right)=\cos \theta$is,$\left\{\frac{\mathrm{\pi }}{6},\frac{\mathrm{\pi }}{2},\frac{5\mathrm{\pi }}{6},\frac{3\mathrm{\pi }}{2}\right\}$.

Step-by-step solution

Step 1 Given equation is,

$\sin \left(2\theta \right)=\cos \theta$.

The double angle formula for $\sin \left(2\theta \right)$is $\sin \left(2\theta \right)=2\sin \theta \cos \theta$.

Substitute the known values in the given equation.

$$\begin{gathered} 2\sin \theta \cos \theta =\cos \theta \\ 2\sin \theta \cos \theta -\cos \theta =0 \\ \cos \theta \left(2\sin \theta -1\right)=0 \\ \cos \theta =0or2\sin \theta -1=0 \\ 2\sin \theta =1 \\ \sin \theta =\frac{1}{2} \end{gathered}$$

Step 2 Solving each equation in the interval [0,2π),

$$\begin{gathered} \theta =\frac{\mathrm{\pi }}{6} \\ \theta =\frac{\mathrm{\pi }}{2} \\ \theta =\frac{5\mathrm{\pi }}{6} \\ \theta =\frac{3\mathrm{\pi }}{2} \end{gathered}$$

Hence the solution set is,$\left\{\frac{\mathrm{\pi }}{6},\frac{\mathrm{\pi }}{2},\frac{5\mathrm{\pi }}{6},\frac{3\mathrm{\pi }}{2}\right\}$.