## How to Find Co-terminal Angles and Reference Angles

Read,5 minutes

### Co-terminal Angles: What Are They?

Angles with the **same** starting side and a **common** terminal side are **co-terminal** angles . Despite having various values, these angles are in a **standard** **position**. They share the same **sides**, are in the same **quadrant,** and have the same **vertices**. The terminal sides coincide at the same angle, whether the angles are rotated **clockwise** or **counterclockwise**. An angle measures a ray's **rotation** around its starting point. The starting side of that angle refers to the original ray, while the terminal side denotes the ray's final position upon rotation.

Think about \(65°\). Due to its terminal side in the **first** **quadrant**, it occupies its standard position there. View the illustration.

- \(65°\) returns to its terminal side at \(425°\), following a complete circle counterclockwise. In the first quadrant, \(425°\) and \(65°\) coincide.
- On a complete clockwise revolution, \(65°\) again reaches its terminal side at \(-295°\). In the first quadrant, \(-295°\) and \(65°\) coincide.

As a result, the angles \(425°\) and \(-295°\) are **co-terminal** of \(65°\).

### The formula for Co-terminal Angles

Whether an angle \((θ)\) is expressed in terms of **degrees** or **radians**, the following formula can help you determine its **co-terminal** angles:

- Degrees: \(θ \ ± \ 360n\)
- Radians: \(θ \ ± \ 2πn\)

In the above formula, \(360n\) means a number that is a **multiple** of \(360\), and \(n\) is an **integer** that shows** how many times** the coordinate plane has been turned.

As a result, we may say that the following angles are all coterminal: \(65°, \ -295°, \ 425°, \ - 655°, \ 785°,\) etc. They only differ by a certain number of whole circles. According to this definition, "two angles are said to be co-terminal if the **difference** between the angles is a **multiple** of \(360°\) (or \(2π\), if the angle is in terms of **radians**)".

**Example1**

Find the positive and negative coterminal angles of \(\frac{3π}{4}\)

**Solution:**

- \(\frac{3π}{4} \ + \ 2π \ = \frac{3π}{4} \ + \ \frac{8π}{4} \ = \ \frac{11π}{4}\)
- \(\frac{3π}{4} \ - \ 2π \ = \frac{3π}{4} \ - \ \frac{8π}{4} \ = \ -\frac{5π}{4}\)

**Example2**

Find the positive and negative co-terminal angles of \(59°\).

**Solution:**

- \(59° \ + \ 360° \ = \ 419°\)
- \(59° \ - \ 360° \ = \ -301°\)

### Positive and Negative Co-terminal Angles

Positive or negative co-terminal angles are also possible. we found that \(419°\) and \(-301°\) are the co-terminal angles of \(59°\). Here,

- \(419°\) is the coterminal angle of \(59°\) and that is
**positive**. - \(-301°\) The
**negative**co-terminal angle of is \(59°\).

\(θ \ ± \ 360n\), where \(n\) takes a **positive** number for **counterclockwise** rotations and a **negative** value for **clockwise** rotations. In order to get positive or negative co-terminal angles, we must select whether to **add** or **subtract** multiples of \(360°\) (or \(2π\)).

### Reference Angles

The reference angle for every angle is the angle between \(0°\) and \(90°\). The **quadrant** of the terminal side determines the reference angle.

### The procedures for determining an angle's reference angle

- First, its
**co-terminal**angle must be calculated, which ranges from \(0°\) to \(360°\). - The coterminal angle's
**quadrant**is then visible. - The reference angle is the same as our given angle if the terminal side is in the
**first quadrant**(\(0°\) to \(90°\)). For instance, if the angle is \(40°\) degrees, the reference angle is \(40°\) degrees. - The reference angle is \(180° \ -\) given angle if the terminal side is in the
**second**quadrant (\(90°\) to \(180°\) degrees). For instance, if the angle is \(170°\), the reference angle is \(180° \ – \ 170° \ = \ 10°\). - The reference angle is (given angle \(- \ 180°\)) if the terminal side is in the
**third**quadrant (\(180°\) to \(270°\)). For instance, if an angle is \(195°\), its reference angle is \(195° \ – \ 180°\), which equals \(15°\). - The reference angle is (\(360° \ -\) provided angle) if the terminal side is in the
**fourth**quadrant (\(270°\) to \(360°\)). For instance, if the angle is \(305°\), the reference angle is \(360° \ – \ 305° \ = \ 55°\).

**Example:**

Find the reference angle of \(510°\).

**Solution:**

Find the **co-terminal** angle of \(510°\). \(510° \ - \ 360° \ = \ 150°\) is the co-terminal angle.

The second **quadrant** contains the **terminal** **side**. The reference angle is thus \(180° \ – \ 150° \ = \ 30°\).

As a result, \(30°\) is the reference angle for \(510°\).

### Summary

- Co-terminal angles are equal angles.

- To find a co-terminal of an angle, add or subtract \(360\) degrees (or \(2π\) for radians) to the given angle.

- Reference angle is the smallest angle that you can make from the terminal side of an angle with the \(x-\)axis.

### Exercises for Finding Co-terminal Angles and Reference Angles

**1) **Find the co-terminal angle: \(220° \ =\)

**2) **Find the co-terminal angle: \(115° \ =\)

**3) **Find the co-terminal angle: \(\frac{5π}{4} \ =\)

**4) **Find the co-terminal angle: \(\frac{13π}{20} \ =\)

**5) **Find the co-terminal angle: \(\frac{9π}{14} \ =\)

**6) **Find the reference angle: \(640° \ =\)

**7) **Find the reference angle: \(580° \ =\)

**8) **Find the reference angle: \(420° \ =\)

**9) **Find the reference angle: \(\frac{19π}{8} \ =\)

**10) **Find the reference angle: \(\frac{27π}{7} \ =\)

**1) **Find the co-terminal angle: \(220° \ =\)

- Positive co-terminal: \(\color{red}{220° \ = \ 220° \ + \ 360° \ = \ 580°}\)
- Negative co-terminal: \(\color{red}{220° \ = \ 220° \ - \ 360° \ = \ -140°}\)

**2) **Find the co-terminal angle: \(115° \ =\)

- Positive co-terminal: \(\color{red}{115° \ = \ 115° \ + \ 360° \ = \ 475°}\)
- Negative co-terminal: \(\color{red}{115° \ = \ 115° \ - \ 360° \ = \ -245°}\)

**3) **Find the co-terminal angle: \(\frac{5π}{4} \ =\)

- Positive co-terminal: \(\color{red}{\frac{5π}{4} \ = \frac{5π}{4} \ + \ 2π \ = \ \frac{13π}{4}}\)
- Negative co-terminal: \(\color{red}{\frac{5π}{4} \ = \frac{5π}{4} \ - \ 2π \ = \ -\frac{3π}{4}}\)

**4) **Find the co-terminal angle: \(\frac{13π}{20} \ =\)

- Positive co-terminal: \(\color{red}{\frac{13π}{20} \ = \frac{13π}{20} \ + \ 2π \ = \ \frac{53π}{20}}\)
- Negative co-terminal: \(\color{red}{\frac{13π}{20} \ = \frac{13π}{20} \ - \ 2π \ = \ -\frac{27π}{4}}\)

**5) **Find the co-terminal angle: \(\frac{9π}{14} \ =\)

- Positive co-terminal: \(\color{red}{\frac{9π}{14} \ = \frac{9π}{14} \ + \ 2π \ = \ \frac{27π}{14}}\)
- Negative co-terminal: \(\color{red}{\frac{9π}{14} \ = \frac{9π}{14} \ - \ 2π \ = \ -\frac{21π}{14}}\)

**6) **Find the reference angle: \(640° \ =\)

Co-terminal angle: \(\color{red}{640° \ = \ 640° \ - \ 360° \ = \ 280° \ ⇒}\) **fourth** quadrant

\(\color{red}{⇒}\) reference angle \(\color{red}{= \ 360° \ - \ 280° \ = \ 80°}\)

**7) **Find the reference angle: \(580° \ =\)

Co-terminal angle: \(\color{red}{580° \ = \ 580° \ - \ 360° \ = \ 220° \ ⇒}\) **third** quadrant

\(\color{red}{⇒}\) reference angle \(\color{red}{= \ 220° \ - \ 180° \ = \ 40°}\)

**8) **Find the reference angle: \(420° \ =\)

Co-terminal angle: \(\color{red}{420° \ = \ 420° \ - \ 360° \ = \ 60° \ ⇒}\) **first** quadrant

\(\color{red}{⇒}\) reference angle \(\color{red}{= \ 60°}\)

**9) **Find the reference angle: \(\frac{19π}{8} \ =\)

Co-terminal angle: \(\color{red}{\frac{14π}{8} \ = \ \frac{19π}{8} \ - \ 2π \ = \ \frac{3π}{8} \ ⇒}\) **first** quadrant

\(\color{red}{⇒}\) reference angle \(\color{red}{= \ \frac{3π}{8}}\)

**10) **Find the reference angle: \(\frac{27π}{7} \ =\)

Co-terminal angle: \(\color{red}{\frac{27π}{7} \ = \ \frac{27π}{7} \ - \ 2π \ = \ \frac{13π}{7} \ ⇒}\) **fourth** quadrant

\(\color{red}{⇒}\) reference angle \(\color{red}{= \ 2π \ - \ \frac{13π}{7} \ = \ \frac{π}{7}}\)

## Finding Co-terminal Angles and Reference Angles Practice Quiz

### More Trigonometric Functions courses

- How to Find the Length of Arc and the Area of Sector
- How to Write Each Measure in Radians
- How to Find Missing Sides and Angles of a Right Triangle
- How to Write Each Measure in Degrees
- How to Evaluate Each Trigonometric Function
- How to Sketch Angles in Standard Position
- How to Find Co-terminal Angles and Reference Angles
- How to Find Trigonometric Ratios of General Angles