Question

How do you find the reference angle for $ - 515$ degrees?

Answer 1

Hint: In this question, we need to find the reference angle for a given data. Finding the reference angle is very simple as adding or subtracting ${360^ \circ }$ or $2\pi $ to the given angle, counting on whether the given angle is in degrees or radians. We add ${360^ \circ }$ until we get a positive angle. Then we find the quadrant in which the resultant angle lies. Then depending on the quadrant, we will find the required reference angle and obtain the result.

Complete step-by-step answer:
In this problem, we are asked to determine the reference angle for $ - 515$ degrees.
Let us first understand what exactly we mean by a reference angle.
The reference angle is defined as the closest angle made with the x-axis regardless of the position where it ends up.
Now let us find out the reference angle for $ - 515$ degrees.
Since the given angle is negative, the reference angle is obtained by simply adding one full rotation (360 degrees) until we get a positive angle.
Now we add a positive rotation i.e. 360 degrees to $ - 515$ degrees, we get,
$ \Rightarrow - 515 + 360 = - 155$
which is a negative angle. So adding again 360 degrees to $ - 155$ degrees we get,
$ \Rightarrow - 155 + 360 = 205$
which is a positive angle.
So now we have a positive angle. Now we find out in which quadrant 205 degrees lies.
We know that the angles in the third quadrant are in the interval ${180^\circ } \leqslant \theta \leqslant {270^\circ }$.
So the angle we obtained which is 205 degrees lies in the third quadrant.
So for the third, we find the reference angle by using the formula,
Reference angle $ = \theta - {180^\circ }$.
Here $\theta = {205^\circ }$. Hence substituting the value of $\theta $, we get,
Reference angle $ = {205^\circ } - {180^\circ }$
$ \Rightarrow $Reference angle $ = {25^\circ }$.
Hence the reference angle for $ - 515$ degrees is ${25^\circ }$.

Note:
Reference angles are the measure between a given angle and the x-axis.
The following steps must be followed to find out the reference angle.
(1) Keep subtracting 360 degrees from the given angle until it lies between ${0^ \circ }$ and ${360^ \circ }$. For negative angles add 360 degrees instead.
(2) Determine in which quadrant the resultant angle $\theta $ lies.
(3) Depending on the quadrant, find the reference angle $\theta $.
For the first quadrant, the reference angle will be the same as $\theta $.
For the second quadrant, the reference angle is ${180^ \circ } - \theta $.
For the third quadrant, the reference angle is $\theta - {180^ \circ }$.
For the fourth quadrant, the reference angle is ${360^ \circ } - \theta $.