Reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis. It is positive and always lies between 0 and 90 degrees (Acute angle). Reference angle can be in any of the four quadrants. When determining the reference angle, vertically drop (raise) the line to the horizontal axis (x-axis). Angles in first quadrant are their own reference angles. The angle is measured by the amount of rotation from the initial side to the terminal side.

## Reference Angle Formula

Let $\theta$ be the given reference angle. Given below is the table to find reference angle in different quadrants:

 Quadrant I $\theta$ Quadrant II 180$^{\circ}$ - $\theta$ Quadrant III $\theta$ - 180$^{\circ}$ Quadrant IV 360$^{\circ}$ - $\theta$

## Reference Angle Chart

To find reference angle in four different quadrants, given below is the chart, where in the figure x-axis is the frame of reference. A is the angle and $A_{r}$ is the reference angle.

While computing the measure (in radians) of the reference angle for any given angle $\theta$, remember to use the following table:

 Quadrant Measure of angle $\theta$ Measure of reference angle | 0 to $\frac{\pi}{2}$ $\theta$ || $\frac{\pi}{2}$ to $\pi$ $\pi$  - $\theta$ ||| $\pi$ to $\frac{3\pi}{2}$ $\theta$ - $\pi$ |V $\frac{3\pi}{2}$ to 2$\pi$ 2$\pi$ - $\theta$

For example, let us find the reference angle for 5$\frac{\pi}{4}$

To find the reference angle for 5$\frac{\pi}{4}$, follow these simple steps.

1. Determine in which quadrant 5$\frac{\pi}{4}$ lies.

5$\frac{\pi}{4}$ is having its terminal side in the third quadrant.

2. Perform the operations indicated for that quadrant.

The reference angle for the third quadrant = 5$\frac{\pi}{4}$ - $\pi$ = $\frac{\pi}{4}$

Therefore, the reference angle is $\frac{\pi}{4}$

## Reference Angle of a Negative Angle

Remember that positive angles are rotated counter-clockwise and negative angles rotate clockwise. Reference angle is always positive, even if it lies in the third or the fourth quadrant. In case if reference angle cannot be found for some angle, then determine the co-terminal angle which will have the same reference angle.

Co-terminal angles are multiples of 360 degrees apart from each other.

As we know, reference angles are found between 0 and 360 degrees. One can probably add or subtract 360$^{\circ}$ until a number lies between 0$^{\circ}$ and 360$^{\circ}$.

### Solved Example

Question: Find the reference angle for -50$^{\circ}$.
Solution:
Adding 360$^{\circ}$ to -50$^{\circ}$, we get A = 310$^{\circ}$

-50$^{\circ}$ terminates in the fourth quadrant.
So, its reference angle is 360$^{\circ}$ - 310$^{\circ}$ = 50$^{\circ}$.
Therefore, the reference angle for -50$^{\circ}$ is 50$^{\circ}$.

## Finding the Reference Angle

Reference angle of a particular angle depends on the quadrant in which it lies. Reference angle always uses x-axis as the frame of reference.

Every positive angle in the first quadrant is acute. So, the reference angle will be the measure of the angle itself.

In the second quadrant, for a given angle say x$^{\circ}$,  reference angle is got by subtracting x$^{\circ}$ from 180$^{\circ}$.

In the third quadrant, for a given angle say x$^{\circ}$, reference angle is got by subtracting 180$^{\circ}$ from x$^{\circ}$.

Reference angle in the fourth quadrant:

In the fourth quadrant, for a given angle say x$^{\circ}$,   reference angle is got by subtracting x$^{\circ}$ from 360$^{\circ}$.

## Reference Angle Problems

Given below are some of the example problems on reference angles.

### Solved Examples

Question 1: Find the reference angle for the following
1. 56$^{\circ}$
2. 110$^{\circ}$
3. 215$^{\circ}$
4. 320$^{\circ}$

Solution:
1. 56$^{\circ}$ lies in first quadrant. So, the reference angle will be the measure of the angle itself.
Therefore, the reference angle = 56$^{\circ}$

2. 110$^{\circ}$ lies in second quadrant and the formula for reference angle in second quadrant is 180$^{\circ}$ - $\theta$.
Here, $\theta$ = 110$^{\circ}$.
So, 180$^{\circ}$ - 110$^{\circ}$ = 70$^{\circ}$
Therefore, the reference angle = 70$^{\circ}$

3. 215$^{\circ}$ lies in third quadrant and the formula for reference angle in third quadrant is $\theta$ - 180$^{\circ}$
Given: $\theta$ = 215$^{\circ}$
So, 215$^{\circ}$ - 180$^{\circ}$ = 35$^{\circ}$
Therefore, the reference angle = 35$^{\circ}$

4. 320$^{\circ}$ lies in fourth quadrant and the formula for reference angle in fourth quadrant is 360$^{\circ}$ - $\theta$.
Given: $\theta$ = 320$^{\circ}$
So, 360$^{\circ}$ - 320$^{\circ}$ = 40$^{\circ}$
Therefore, the reference angle = 40$^{\circ}$.

Question 2: What is the reference angle for -280$^{\circ}$?
Solution:
We add 360$^{\circ}$ to -280$^{\circ}$
$\Rightarrow$ 360$^{\circ}$ - 280$^{\circ}$ = 80$^{\circ}$
80$^{\circ}$ lies in first quadrant as it is in between 0$^{\circ}$ and 90$^{\circ}$. They are their own reference angles.
Therefore, the reference angle is 80$^{\circ}$.