Question

In Problems 11–34, solve each equation on the interval $0\le \theta \le 2\pi$.

$1-\cos \theta =\frac{1}{2}$

Short answer

The solution set is$\left\{\frac{\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3}\right\}$

Step-by-step solution

Step 1. Given Information

In the given problem we have to solve each equation on the interval$0\le \theta \le 2\pi$.

$1-\cos \theta =\frac{1}{2}$

Step 2. Firstly solving the equation 1-cosθ=12

Subtract 1 on both side

$$\begin{gathered} 1-\cos \theta -1=\frac{1}{2}-1 \\ -\cos \theta =\frac{1}{2}-1·\frac{2}{2} \\ -\cos \theta =\frac{1}{2}-\frac{2}{2} \\ -\cos \theta =\frac{1-2}{2} \\ -\cos \theta =\frac{-1}{2} \end{gathered}$$

Multiply with $-1$on both side

role="math" localid="1646565140078" $$\begin{gathered} -\cos \theta \times (-1)=\frac{-1}{2}\times (-1) \\ \cos \theta =\frac{1}{2} \end{gathered}$$

Step 3. After solving the equation cosθ=12

The period of the cosine function is $2\mathrm{\pi }$. In the interval $[0,2\pi )$, there are two angles $\theta$for whichlocalid="1646566828893" $\cos \theta =\frac{1}{2}:\theta =\frac{\pi }{3}\mathrm{and}\theta =\frac{5\pi }{3}$

So the solutions set is$\frac{\mathrm{\pi }}{3},\frac{5\mathrm{\pi }}{3}$