Question

1. let $\cos(60)=\frac{1}{2}$. find the angle measure $(\theta)$ in degrees, for $\sin(\theta)=\frac{1}{2}$.

Explanation:

Step1: Recall the sine function values

We know that for angles in the range of \(0^{\circ}\leq\theta\leq360^{\circ}\), \(\sin\theta=\frac{1}{2}\) has solutions.
The basic angle for which \(\sin\alpha=\frac{1}{2}\) is \(\alpha = 30^{\circ}\) (from the unit - circle or special - angle values: \(\sin30^{\circ}=\frac{1}{2}\)). Also, using the identity \(\sin(180^{\circ}-\alpha)=\sin\alpha\), when \(\alpha = 30^{\circ}\), \(\sin(180 - 30)^{\circ}=\sin150^{\circ}=\frac{1}{2}\)

Step2: Determine the general solution

The general solution of the equation \(\sin\theta=\sin\alpha\) is \(\theta = n\times360^{\circ}+\alpha\) or \(\theta=n\times360^{\circ}+(180^{\circ}-\alpha)\), where \(n\in\mathbb{Z}\).
For \(n = 0\), \(\theta = 30^{\circ}\) or \(\theta=150^{\circ}\)

Answer:

\(\theta = 30^{\circ}\) or \(\theta = 150^{\circ}\)