Chapter 6.4, Problem 22E

To determine

To Find: All the angles $\theta$ between $0^0$ and ${180}^0$ which satisfied the mentioned equation in the problem.

Answer to Problem 22E

All the angles $\theta$ are ${60}^0$ and ${120}^0$ .

Explanation of Solution

Given information:

The equation mentioned in the problem is $\sin \theta =\frac{\sqrt{3}}{2}$ .

Formula:

The trigonometric formula $\sin \left(\pi -\theta \right)=\sin \theta$ .

Calculation:

Consider the statement equation mentioned in the problem is $\sin \theta =\frac{\sqrt{3}}{2}$ .

Since $\sin \left(\theta \right)=\frac{\sqrt{3}}{2}=\sin {60}^0\Rightarrow \theta ={60}^0$ .

Recall that $\sin \left(\pi -\theta \right)=\sin \theta$ .

Therefore $\sin \left({60}^0\right)=\sin \left({180}^0-{120}^0\right)=\sin \left({120}^0\right)$ .

Therefore all the angles between $\theta$ are ${60}^0$ and ${120}^0$ .