Question

evaluate without a calculator: csc 210°

Explanation:

Step1: Find the reference angle

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

Step2: Determine the sign

\(210^{\circ}\) is in the third quadrant. In the third quadrant, \(\sin\theta\lt0\), and since \(\csc\theta=\frac{1}{\sin\theta}\), \(\csc\theta\lt0\) in the third quadrant.

Step3: Use the reciprocal identity

We know that \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin30^{\circ}=\frac{1}{2}\). So \(\csc210^{\circ}=\frac{1}{\sin210^{\circ}}\). Substituting \(\sin210^{\circ}=-\sin30^{\circ}=-\frac{1}{2}\), we get \(\csc210^{\circ}=\frac{1}{-\frac{1}{2}}\).

Answer:

\(- 2\)