Question

1. state the reference angle in both radians and degrees: 330°
2. name an angle x that is coterminal to 220°
3. determine the exact trig ratio: cos(3π/4)
4. determine the exact trig ratio: csc240°

Explanation:

Step1: Find the reference angle for \(330^{\circ}\)

Since \(330^{\circ}\) is in the fourth - quadrant (\(270^{\circ}<330^{\circ}<360^{\circ}\)), the reference angle \(\theta_{r}=360^{\circ}-330^{\circ}=30^{\circ}\). To convert the reference angle from degrees to radians, use the formula \(\theta_{r}^{rad}=\theta_{r}^{deg}\times\frac{\pi}{180}\). So, \(30^{\circ}\times\frac{\pi}{180}=\frac{\pi}{6}\) radians.

Step2: Determine \(\cos\frac{3\pi}{4}\)

The angle \(\frac{3\pi}{4}\) is in the second - quadrant (\(\frac{\pi}{2}<\frac{3\pi}{4}<\pi\)). The reference angle \(\theta_{r}=\pi - \frac{3\pi}{4}=\frac{\pi}{4}\). We know that \(\cos\theta=-\cos\theta_{r}\) for \(\theta\) in the second - quadrant. And \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), so \(\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}\).

Step3: Determine \(\csc240^{\circ}\)

The angle \(240^{\circ}\) is in the third - quadrant (\(180^{\circ}<240^{\circ}<270^{\circ}\)). The reference angle \(\theta_{r}=240^{\circ}-180^{\circ}=60^{\circ}\). We know that \(\csc\theta=-\csc\theta_{r}\) for \(\theta\) in the third - quadrant. Since \(\csc\theta=\frac{1}{\sin\theta}\) and \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), then \(\csc60^{\circ}=\frac{2\sqrt{3}}{3}\), so \(\csc240^{\circ}=-\frac{2\sqrt{3}}{3}\).

Step4: Find a coterminal angle to \(220^{\circ}\)

Coterminal angles are given by \(\theta + n\times360^{\circ}\), where \(n\in\mathbb{Z}\). If \(n = 1\), then \(x=220^{\circ}+360^{\circ}=580^{\circ}\) (we can also take \(n=-1\) and get \(x = 220^{\circ}-360^{\circ}=-140^{\circ}\), but \(580^{\circ}\) is a positive example).

Answer:

1. Reference angle: \(30^{\circ}\) or \(\frac{\pi}{6}\) radians.
2. An angle \(x\) coterminal to \(220^{\circ}\): \(580^{\circ}\) (for example).
3. \(\cos\frac{3\pi}{4}=-\frac{\sqrt{2}}{2}\).
4. \(\csc240^{\circ}=-\frac{2\sqrt{3}}{3}\).