Question

find the exact value without a calculator: csc 330° = ?

Explanation:

Step1: Find the reference angle

Since \(330^{\circ}=360^{\circ} - 30^{\circ}\), the reference angle is \(30^{\circ}\).

Step2: Determine the sign

\(330^{\circ}\) is in the fourth quadrant. In the fourth quadrant, \(\sin\theta\) is negative, and \(\csc\theta=\frac{1}{\sin\theta}\), so \(\csc\theta\) is negative.

Step3: Calculate \(\csc330^{\circ}\)

We know that \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin30^{\circ}=\frac{1}{2}\). So \(\csc330^{\circ}=\frac{1}{\sin(360^{\circ} - 30^{\circ})}\).
Since \(\sin(A - B)=\sin A\cos B-\cos A\sin B\), here \(A = 360^{\circ}\), \(B=30^{\circ}\), \(\sin(360^{\circ}-30^{\circ})=\sin360^{\circ}\cos30^{\circ}-\cos360^{\circ}\sin30^{\circ}=0\times\frac{\sqrt{3}}{2}-1\times\frac{1}{2}=-\frac{1}{2}\).
Then \(\csc330^{\circ}=\frac{1}{-\frac{1}{2}}=- 2\).

Answer:

\(-2\)