Question

How do you determine the two coterminal angles (one positive one negative) for the angle ${45^ \circ }$?

Answer 1

Hint: This problem deals with finding the coterminal angles of a given angle. Coterminal angles are angles who share the terminal sides. Finding coterminal angles is as simple as adding or subtracting ${360^ \circ }$ or $2\pi $ to each angle, depending on whether the given angle is in degrees or radians. The coterminal angles can be positive or negative.

Complete step-by-step solution:
To find the coterminal angles, simply add or subtract ${360^ \circ }$ degrees as many times as needed from the reference angle.
We are given with an angle of ${45^ \circ }$.
Now we have to find the coterminal angles for the given angle ${45^ \circ }$.
To find the positive coterminal angle of ${45^ \circ }$, we have to add the angle ${360^ \circ }$ to it, as given below:
\[ \Rightarrow {45^ \circ } + {360^ \circ } = {405^ \circ }\]
So the positive coterminal angle of ${45^ \circ }$ is \[{405^ \circ }\].
Now to find the negative coterminal angle of ${45^ \circ }$, we have to subtract the angle ${360^ \circ }$ from the angle ${45^ \circ }$, as given below:
\[ \Rightarrow {45^ \circ } - {360^ \circ } = - {315^ \circ }\]
So the negative coterminal angle of ${45^ \circ }$ is \[ - {315^ \circ }\].

The positive coterminal angle and the negative coterminal of ${45^ \circ }$ are \[{405^ \circ }\] and \[ - {315^ \circ }\] respectively.

Note: Please note that while finding the coterminal angles of a given angle. If we given an angle of \[{500^ \circ }\], and we are asked to find the positive coterminal angle and negative coterminal angle of the angle \[{500^ \circ }\], then first we have to subtract the angle ${360^ \circ }$ from the angle \[{500^ \circ }\], which results in the angle of \[{140^ \circ }\] now add and subtract the angle ${360^ \circ }$ from the angle \[{140^ \circ }\], where the positive coterminal is the same as the angle \[{500^ \circ }\].