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)