Question

find the reference angle for the given angle.

1. - 780°

a) 240°
b) 120°
c) 60°
d) - 60°
17)\frac{-\pi}{4}
a)\frac{3\pi}{4}
b)\frac{7\pi}{4}
c)\frac{\pi}{4}
d)\frac{-\pi}{4}

Explanation:

Question 16

Step1: Find the coterminal angle

To find a coterminal angle of \(-780^{\circ}\), we add \(360^{\circ}\) multiple times.
\(-780^{\circ}+3\times360^{\circ}=-780^{\circ} + 1080^{\circ}=300^{\circ}\)

Step2: Determine the reference angle

Since \(300^{\circ}\) is in the fourth - quadrant (\(270^{\circ}<300^{\circ}<360^{\circ}\)), the reference angle formula for an angle \(\theta\) in the fourth - quadrant is \(360^{\circ}-\theta\).
For \(\theta = 300^{\circ}\), the reference angle is \(360^{\circ}-300^{\circ}=60^{\circ}\)

Step1: Analyze the angle \(-\frac{\pi}{4}\)

The angle \(-\frac{\pi}{4}\) is equivalent to an angle of \(2\pi-\frac{\pi}{4}=\frac{8\pi - \pi}{4}=\frac{7\pi}{4}\) (by adding \(2\pi\) to get a positive coterminal angle).
Since \(\frac{7\pi}{4}\) is in the fourth - quadrant (\(\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi\)), and for an angle \(\theta=\frac{7\pi}{4}\) in the fourth - quadrant, the reference angle formula is \(2\pi-\theta\).
\(2\pi-\frac{7\pi}{4}=\frac{8\pi - 7\pi}{4}=\frac{\pi}{4}\)

Answer:

C. \(60^{\circ}\)

Question 17