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Reference Angle

In trigonometry, we learn about four quadrants and angles appearing in those quadrants. Reference angle is a concept that is quite frequently used in context with angles. Reference angle is defined as the positive acute angle that represents an angle of any measure. An angle on the X-Y plane has the reference angle lying is always between 0 degree to 90 degree. The reference angle is a smallest angle that has been made from the terminal side of the angle ( the point where angle ends at the X axis). Reference angle may appear in any of the four quadrants. The reference angle in the first quadrant is different, since in the first quadrant, there are reference angle of their own angles. To find reference angle, we use X axis as its frame of reference. If A is a standard angle, then the reference angle of A is represented by $A_{r}$ as shown in the figure given below.

Reference Angle

Different types of reference angles in context with all four quadrants are shown above. Have a look at the reference angle in the first quadrant, the way of whose measurement is little different than second quadrant reference angle. In the first quadrant, it is measured from the positive X axis; while in second quadrant, it is calculated towards the negative X axis. The same similarly could be found in third and fourth quadrant reference angle.

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Reference Angle Definition

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The reference angle for any angle ($\theta$) in standard position is the positive acute angle between the terminal side of angle and the x-axis. Sometime reference angle also called as related angle.

Reference angle of any angle is a acute angle, i.e., reference angle is always between 0$^o$ and 90$^o$.

Reference Angle Formula

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Reference angle is measured on basis of position of the given angle in any of the 4 quadrants in a rectangular plane.

Reference Angle Chart


Quadrant Reference Angle for A (in degrees)
Reference Angle for A (in radians)
1 Same as A
Same as A
2 180$^o$ - A
$\pi$ - A
3 A - 180$^o$ A - $\pi$
4 360$^o$ - A 2$\pi$ - A



Reference Angle Radians

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In case of radians, recall that 360$^o$ is equal to 2$\pi$ radians, and 180° is equal to $\pi$ radians. When radian measure is used, the reference angle is sometimes referred to the reference number.

Let us calculate the reference angle for $\frac{2 \pi}{3}$ ?

Let A = $\frac{2 \pi}{3}$ i.e. 120$^o$ (lies in II quadrant)

In II quadrant, reference Angle for A is $\pi$ - A.

So, we would subtract the value of A from "$\pi$" to get reference angle:

A$_r$ = $\pi$ - $\frac{2 \pi}{3}$ = $\frac{ \pi}{3}$

Thus, reference angle of $\frac{2 \pi}{3}$ is $\frac{ \pi}{3}$.

Reference Angles Unit Circle

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Unit circle contains various values of the angles. It is also helpful to find the reference angle of any given angle.
Reference Angles Unit Circle Diagram

Reference Angle of a Negative Angle

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If we can't find the reference angle for any negative angle, then firstly we have to find the co-terminal angle of that angle. And, co-terminal angles are multiples of 360 degrees apart from each other. So, we can add (or subtract) multiples of 360 to change the original angle into a co-terminal.

Let us find the reference angle for -$\frac{9 \pi}{4}$.

Let A = -$\frac{9 \pi}{4}$

To find the co-terminal, add 2$\pi$ in A.

-$\frac{9 \pi}{4}$ is co-terminal with -$\frac{9 \pi}{4}$ + 2$\pi$ = - $\frac{\pi}{4}$

Example on Reference Angle of a Negative Angle

We find that, the reference angle of A is Ar = $\frac{\pi}{4}$.

Finding Reference Angles

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There are four different formulas to find the reference angle of a particular angle, based on the quadrant in which the given angle is present.

Finding Reference angle in First Quadrant (0$^o$ to 90$^o$)

In the first Quadrant, the Reference angle is the angle itself.

Reference angle Ar = A

If angle given is 60 degree, then it's reference angle will be equal to the angle itself.

Reference angle = 60$^o$

Reference Angle in First Quadrant

Finding Reference angle in Second Quadrant (90$^o$ to 180$^o$)

In the second quadrant, the Reference angle is got by subtracting the given angle from 180$^o$.

Reference angle Ar = 180$^o$ - A

If the angle given is 120 degree then the Reference angle is given as

Reference angle = 180$^o$ - 120$^o$ = 60$^o$

Reference Angle in Second Quadrant

Finding Reference angle in Third Quadrant (180$^o$ - 270$^o$)

In the third quadrant, the Reference angle is got by subtracting 180$^o$ from the given angle.

Reference angle Ar = A - 180$^o$

If the angle given is 240 degree then the Reference angle is given as

Reference angle = 240$^o$ - 180$^o$ = 60$^o$

Reference Angle in Third Quadrant

Finding Reference angle in Fourth Quadrant (270$^o$- 360$^o$)

In the fourth quadrant, the Reference angle is got by subtracting the given angle from 360$^o$.

Reference angle Ar = 360$^o$ - A

If the angle given is 315 degree, then the Reference angle is given as

Reference angle = 360$^o$ - 315$^o$ = 45$^o$

Reference Angle in Fourth Quadrant

Reference Angles Example Problems

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

Solved Examples

Question 1: Find the reference angle for the graph given below:

Example on reference angles


Solution:

As we know, reference angle always use x axis as the frame of reference. From the figure shown above, we can see that the given angle lies in the second quadrant. So, the Reference angle is given by the formula,

Reference angle = 180$^o$ - Given angle

Reference angle for 135$^o$ is given as = 180$^o$ - 135$^o$ = 45$^o$

Example for reference angles



Question 2:

Measure the Reference angle for the following angles.

  1. 49$^o$
  2. 108$^o$
  3. 345$^o$

Solution:

1) As we can see, 49$^o$ lies in the first quadrant. So, Reference angle is given by the formula,

Reference angle Ar = A( given angle)

Hence, the Reference angle = 49$^o$

2) As we can see, 108$^o$ lies in second quadrant. So, Reference angle is given by the formula,

Reference angle Ar = 180$^o$ - A

Hence, the Reference angle = 72$^o$

3) Angle given is 345$^o$. As we can see, this angle lies in the fourth quadrant. So, reference angle is given by the formula,

Reference angle Ar = 360$^o$ - A

Hence, the Reference angle = 15$^o$.



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