Question

evaluate sin(210°) exactly. note: enter the exact, fully simplified answer.

Explanation:

Step1: Determine the reference angle

Since \(210^{\circ}=180^{\circ} + 30^{\circ}\), the reference angle \(\theta_{r}=210^{\circ}-180^{\circ}=30^{\circ}\)

Step2: Determine the sign

The angle \(210^{\circ}\) is in the third quadrant. In the third quadrant, \(\sin\theta<0\)

Step3: Use the reference - angle formula

We know that \(\sin\theta=-\sin\theta_{r}\) (for \(\theta\) in the third quadrant). Since \(\theta = 210^{\circ}\) and \(\theta_{r}=30^{\circ}\), and \(\sin30^{\circ}=\frac{1}{2}\)
So, \(\sin(210^{\circ})=-\sin(30^{\circ})\)

Answer:

\(-\frac{1}{2}\)