Question

How do you find an angle with a positive measure and an angle with a negative measure that are coterminal with ${620^ \circ }$?

Answer 1

Hint: In this question, we need to find angles with positive and negative measure that are coterminal to ${620^ \circ }$. coterminal angles are angles who share the identical initial side and terminal sides. Finding coterminal angles is very simple as adding or subtracting ${360^ \circ }$ or $2\pi $ to every angle, counting on whether the given angle is in degrees or radians. We add and subtract ${360^ \circ }$ until we get a positive angle or negative angle respectively.

Complete step by step solution:
Given the measure of the angle ${620^ \circ }$
We are asked to find out an angle with a positive measure and an angle with a negative measure that are coterminal to ${620^ \circ }$.
Let us first define what is coterminal angle.
coterminal angles are those who share the same initial and terminal sides.
It is obtained by simply adding and subtracting one full rotation $({360^ \circ })$ until we get a positive angle or negative angle.
The formula for finding the coterminal angles of a given angle ‘x’ is given by,
${(x + 360n)^\circ }$, where n is any integer.
In the question we have, $x = {620^ \circ }$.
Substituting this value in the formula we get,
$ \Rightarrow {(620 + 360n)^\circ }$
Now we put different values of n to get positive and negative angles.
For $n = 1$, we have,
$ \Rightarrow {(620 + 360 \times 1)^\circ }$
$ \Rightarrow {(620 + 360)^\circ }$
$ \Rightarrow {980^ \circ }$
which is one of positive coterminal angles.
For $n = - 1$, we have,
$ \Rightarrow {\left( {620 + 360 \times ( - 1)} \right)^\circ }$
$ \Rightarrow {(620 - 360)^\circ }$
$ \Rightarrow {260^ \circ }$
which is a positive angle.
For $n = - 2$, we have,
$ \Rightarrow {\left( {620 + 360 \times ( - 2)} \right)^\circ }$
$ \Rightarrow {(620 - 720)^\circ }$
$ \Rightarrow - {100^\circ }$
which is one of negative coterminal angles.

Hence the coterminal angles to ${620^ \circ }$ with a positive measure is ${980^ \circ }$ and with a negative measure is $ - {100^ \circ }$.

Note: We say that an angle is in standard position within the coordinate plane its vertex is found at the origin and one ray is on the positive x-axis. The ray on the x-axis is termed as the initial side and therefore the other ray is termed as the terminal side.
If two angles in standard position have the identical terminal side, then they are called coterminal angles.
The formula for finding the coterminal angles of a given angle ‘x’ is given by,
${(x + 360n)^\circ }$, where n is any integer.
Students must remember the above formula to find out the coterminal angles with a positive and negative measure.