Scroll to Top

Virtual Math Learning Center

Virtual Math Learning Center Texas A&M University Virtual Math Learning Center
tags:
Angle Coterminal Angles Reference Angle Standard Position for Angles Trigonometry Series

Coterminal and Reference Angles Exercise 5

Author: Hannah Solomon

The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Reveal Answer" to see the answer to the problem.

Problem: Find an angle between \(0^\circ\) and \(360^\circ\) that is coterminal to \(-660^\circ\). Then draw the angle in standard position and find its reference angle.

Answer: The Coterminal and Reference Angle are both \(60^\circ.\)

Solution Method: First, we need to find a coterminal angle between 0 and \(360^\circ.\) We see that \(-660^\circ\) is less than 0 so we need to add a \(360^\circ\) rotation. 
\begin{align*}
    -660^\circ + 360^\circ = -300^\circ
\end{align*}
But this measure is still less than 0 so while \(-300^\circ\)is coterminal to \(-660^\circ,\) it's not in the range we want. So we'll add \(360^\circ\) again.
\begin{align*}
    -300^\circ + 360^\circ = 60^\circ
\end{align*}
And, indeed, \(60^\circ\) is between \(0^\circ\) and \(360^\circ.\) 

Now to draw the angle in standard position. As a reminder, an angle is in standard position when the vertex is at the origin and the initial side is the positive x-axis. So the angle in standard position is then determined by its terminal side. \(60^\circ\) is positive so the angle will be measured counterclockwise from the initial side. To determine which quadrant \(60^\circ\) is in we need to think about which quadrantal angles it lies between, \(0^\circ,\) \(90^\circ,\) \(180^\circ,\) \(270^\circ,\) or \(360^\circ.\) So \(60^\circ\) is in Q1. 

We also could have found this terminal side by going clockwise \(300^\circ\) because they are coterminal. 

Now to find the reference angle. Recall, the reference angle is the smallest positive acute angle formed by the x-axis and the terminal side of the angle in standard position. But our terminal side for \(60^\circ\) lies in Q1 and is acute so it is the reference angle. 
 

See more videos from this section

Prev Video Return to Section