# Reference Angle

Ava was asked to find the value of \(\sin 45^\circ\) and \(\sin 135^\circ\). She uses the trigonometric table shown below to determine its value as \(\dfrac{\sqrt 2}{2}\).

She finds it hard to solve this problem because the value of 135^{o} is not present in the table. Her friend explains to her that this can be solved using the reference angle of 135^{o}

But do you know what is a reference angle? Let's learn all about it in this short lesson.

**Lesson Plan**

**What Is a Reference Angle?**

The **reference angle** is the smallest possible angle made by the terminal side of the given angle with the x-axis.

**How to Draw Reference Angle?**

To draw the reference angle for an angle, identify its terminal side and see by what angle the terminal side is close to the x-axis.

The reference angle of 135^{o} is drawn below:

Here, 45^{o} is the reference angle of 135^{o}

- The reference angle of an angle is always non-negative i.e., a negative reference angle doesn't exist.
- The reference angle of any angle always lies between 0 and \(\dfrac{\pi}{2}\) (both inclusive).

**What Are the Rules for Reference Angles in Each Quadrant?**

Here are the reference angle formulas depending on the quadrant of the given angle.

Quadrant |
Angle, \(\theta\) | Reference angle |
---|---|---|

I | lies between 0^{o} and 90^{o} |
\(\theta\) |

II | lies between 90^{o} and 180^{o} |
\(180-\theta\) |

III | lies between 180^{o} and 270^{o} |
\(\theta-180\) |

IV | lies between 270^{o} and 360^{o} |
\(360-\theta\) |

If the angle is in radians, then we use the same rules above by replacing 180^{o} with \(\pi\) and 360^{o} with \(2\pi\).

**Example**

Find the reference angle of 135^{o}.

**Solution**

The given angle is, \(\theta=135^\circ\)

We know that 135^{o} lies in quadrant II.

Using the above rules, its reference angle is,

\[180-\theta=180-135 = 45^\circ\]

Therefore, the reference angle of 135^{o} is **45 ^{o}.**

**How to Find Reference Angles?**

In the previous section, we learned that we could find the reference angles using the set of rules mentioned in the table.

That table works only when the given angle lies between 0^{o} and 360^{o}.

But what if the given angle does not lie in this range?

Let's see how we can find the reference angles in this case.

**Steps to Find the Reference Angles**

The steps to find the reference angle of an angle is explained with an example.

**Example **

Find the reference angle of 480^{o}.

**Solution**

**Step 1: Find the coterminal angle of the given angle that lies between 0 ^{o} and 360^{o}.**

The coterminal angle can be found either by adding or subtracting 360^{o} from the given angle as many times as required.

Let's find the coterminal angle of 480^{o} that lies between 0^{o} and 360^{o}.

We will subtract 360^{o} from 480^{o} to find its coterminal angle.

\[480^\circ - 360^\circ = 120^\circ\]

**Step 2: If the angle from step 1 lies between 0 ^{o} and 90^{o}, then that angle itself is the reference angle of the given angle. **

Else, we have to check whether it is closest to 180^{o} or 360^{o} and by how much**.**

Here, 120^{o} does not lie between 0^{o} and 90^{o}.

Also, 120^{o} is closest to 180^{o} by 60^{o}. i.e.,

\[180^\circ - 120^\circ = 60^\circ\]

**Step 3: ****The angle from step 2 is the reference angle of the given angle.**

Thus, the reference angle of 480^{o} is **60 ^{o}.**

- We use the reference angle to find the values of trigonometric functions at an angle that is beyond 90
^{o}.

For example, we can see that the coterminal angle and reference angle of 495^{o}are 135^{o}and 45^{o}respectively.\( \begin{align}&\sin 495^\circ \\[0.2cm] &= \sin 135^\circ \\[0.2cm] &= +\sin 45^\circ \end{align}\)

We have included the + sign because 135

^{o}is in quadrant II, where sine is positive.\(\begin{align}\sin 495^\circ = \dfrac{\sqrt 2}{2}\,\,\, \text{ [Using unit circle]} \end{align}\)

- If we use reference angles, we don't need to remember the complete unit circle, instead we can just remember the first quadrant values of the unit circle.

**Reference Angle Calculator**

Here is the reference angle calculator.

You can enter any angle here and it will give the corresponding reference angle with step-by-step explanations.

**Solved Examples**

Example 1 |

Find the reference angle of \(\dfrac{8\pi}{3}\)

**Solution**

The given angle is greater than \(2\pi\)

**Finding coterminal angle**

We find its coterminal angle by subtracting \(2\pi\) from it.

\[\dfrac{8\pi}{3} - 2\pi = \dfrac{2\pi}{3}\]

This angle does not lie between 0 and \(\dfrac{\pi}{2}\). Hence, it is not the reference angle of the given angle.

**Finding reference angle**

Let's check whether \(\dfrac{2\pi}{3}\) is close to \(\pi\) or \(2\pi\) and by how much.

Clearly, \(\dfrac{2\pi}{3}\) is close to \(\pi\) by \[\pi-\dfrac{2\pi}{3} = \dfrac{\pi}{3}\]

Therefore,

\(\therefore\) The reference angle of \(\dfrac{8\pi}{3}\) is \(\dfrac{\pi}{3}\) |

Example 2 |

Using the above example (Example 1), find \(\cos \dfrac{8\pi}{3}\).

**Solution**

In the above example, we found that the coterminal angle of \(\dfrac{8\pi}{3}\) is \(\dfrac{2\pi}{3}\) which lies in quadrant II, where cos is negative.

\[\cos \dfrac{8\pi}{3} = - \cos \dfrac{2\pi}{3}\]

Also, we found that the reference angle of \(\dfrac{8\pi}{3}\) is \(\dfrac{\pi}{3}\). So

\[ \begin{align}\cos \dfrac{8\pi}{3} &= - \cos \dfrac{2\pi}{3}\\[0.2cm]

&= - \cos \dfrac{\pi}{3}\\[0.2cm]

&= - \dfrac{1}{2}\,\,\,\,\,\text{ (Using the trigonometry table)} \end{align} \]

Therefore,

\(\therefore\)\(\cos \dfrac{8\pi}{3} = - \dfrac{1}{2}\) |

Example 3 |

Find the reference angle of -1200^{o}.

**Solution**

The given angle is -1200^{o}.

**Finding the coterminal angle**

We see that \[ \dfrac{1200}{360}= 3.333...\]

This indicates that there are 3 complete angles (360^{o} angles) in 1200^{o}.

To find a positive coterminal angle of -1200^{o}, we need to add 4 multiples of 360^{o} to it.

(If we add just 3 multiples of 360^{o} to it, the answer won't be a positive angle).

\[-1200^\circ + 4(360^\circ) = 240^\circ\]

**Finding the reference angle**

Let's check whether 240^{o} is close to 180^{o} or 360^{o} and by how much.

Clearly, 240^{o }is close to 180^{o} and by 60^{o}

Therefore,

\(\therefore\) The reference angle of -1200^{o} is 60^{o} |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about the reference angle with the simulations and practice questions. Now, you will be able to easily solve problems on reference angle formula, reference angle calculator, negative reference angle, and how to draw reference angle.

**About Cuemath**

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**Frequently Asked Questions (FAQs)**

## 1. What is the reference angle for a 200^{o} angle?

Between the angles 180^{o} and 360^{o}, we can say that 200^{o} is close to 180^{o} by 20^{o}

Thus, the reference angle of 200^{o} is 20^{o}

## 2. What is the reference angle for a 275^{o} angle?

Between the angles 180^{o} and 360^{o}, we can say that 275^{o} is close to 360^{o} by 85^{o}

Thus, the reference angle of 275^{o} is 85^{o}

## 3. What is the reference angle for a 235^{o} angle?

Between the angles 180^{o} and 360^{o}, we can say that 235^{o} is close to 180^{o} by 55^{o}.

Thus, the reference angle of 235^{o} is 55^{o}