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.

Free printable Worksheets

Exercises for Finding Co-terminal Angles and Reference Angles

1) For \(40°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

2) For \(135°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

3) For \(250°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

4) For \(330°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

5) For \(-45°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

6) For \(405°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

7) For \(-200°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

8) For \(765°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

9) For \(\frac{5π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

10) For \(\frac{7π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

11) For \(-\frac{2π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

12) For \(\frac{11π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

13) For \(\frac{13π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

14) For \(-\frac{17π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

15) For \(\frac{19π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

16) For \(-\frac{25π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

17) For \(1120°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

18) For \(-940°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

19) For \(\frac{29π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

20) For \(-\frac{37π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Answers

1)For \(40°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle already lies between \(0°\) and \(360°\). Subtract \(360°\) to get a negative coterminal angle.

One positive coterminal angle is

\(40°\)

, and one negative coterminal angle is

\(-320°\)

The reference angle is

\(\color{red}{40°}\)

2)For \(135°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant II. Subtract \(360°\): \(135°-360°=-225°\).

One positive coterminal angle is

\(135°\)

, and one negative coterminal angle is

\(-225°\)

The reference angle is

\(\color{red}{45°}\)

3)For \(250°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant III. Subtract \(360°\): \(250°-360°=-110°\).

One positive coterminal angle is

\(250°\)

, and one negative coterminal angle is

\(-110°\)

The reference angle is

\(\color{red}{70°}\)

4)For \(330°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant IV. Subtract \(360°\): \(330°-360°=-30°\).

One positive coterminal angle is

\(330°\)

, and one negative coterminal angle is

\(-30°\)

The reference angle is

\(\color{red}{30°}\)

5)For \(-45°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(360°\): \(-45°+360°=315°\).

One positive coterminal angle is

\(315°\)

, and one negative coterminal angle is

\(-45°\)

The reference angle is

\(\color{red}{45°}\)

6)For \(405°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(360°\): \(405°-360°=45°\). Then subtract \(360°\) again for a negative angle.

One positive coterminal angle is

\(45°\)

, and one negative coterminal angle is

\(-315°\)

The reference angle is

\(\color{red}{45°}\)

7)For \(-200°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(360°\): \(-200°+360°=160°\), which is in Quadrant II.

One positive coterminal angle is

\(160°\)

, and one negative coterminal angle is

\(-200°\)

The reference angle is

\(\color{red}{20°}\)

8)For \(765°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(720°\): \(765°-720°=45°\). Then subtract \(360°\) to get a negative coterminal angle.

One positive coterminal angle is

\(45°\)

, and one negative coterminal angle is

\(-315°\)

The reference angle is

\(\color{red}{45°}\)

9)For \(\frac{5π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant II. Subtract \(2π=\frac{12π}{6}\).

One positive coterminal angle is

\(\frac{5π}{6}\)

, and one negative coterminal angle is

\(-\frac{7π}{6}\)

The reference angle is

\(\color{red}{\frac{π}{6}}\)

10)For \(\frac{7π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant IV. Subtract \(2π=\frac{8π}{4}\).

One positive coterminal angle is

\(\frac{7π}{4}\)

, and one negative coterminal angle is

\(-\frac{π}{4}\)

The reference angle is

\(\color{red}{\frac{π}{4}}\)

11)For \(-\frac{2π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(2π=\frac{6π}{3}\): \(-\frac{2π}{3}+\frac{6π}{3}=\frac{4π}{3}\).

One positive coterminal angle is

\(\frac{4π}{3}\)

, and one negative coterminal angle is

\(-\frac{2π}{3}\)

The reference angle is

\(\color{red}{\frac{π}{3}}\)

12)For \(\frac{11π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

The angle is in Quadrant IV. Subtract \(2π=\frac{12π}{6}\).

One positive coterminal angle is

\(\frac{11π}{6}\)

, and one negative coterminal angle is

\(-\frac{π}{6}\)

The reference angle is

\(\color{red}{\frac{π}{6}}\)

13)For \(\frac{13π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(4π=\frac{12π}{3}\): \(\frac{13π}{3}-\frac{12π}{3}=\frac{π}{3}\).

One positive coterminal angle is

\(\frac{π}{3}\)

, and one negative coterminal angle is

\(-\frac{5π}{3}\)

The reference angle is

\(\color{red}{\frac{π}{3}}\)

14)For \(-\frac{17π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(2π\) once to get \(-\frac{5π}{6}\), then add \(2π\) again to get \(\frac{7π}{6}\).

One positive coterminal angle is

\(\frac{7π}{6}\)

, and one negative coterminal angle is

\(-\frac{5π}{6}\)

The reference angle is

\(\color{red}{\frac{π}{6}}\)

15)For \(\frac{19π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(4π=\frac{16π}{4}\): \(\frac{19π}{4}-\frac{16π}{4}=\frac{3π}{4}\).

One positive coterminal angle is

\(\frac{3π}{4}\)

, and one negative coterminal angle is

\(-\frac{5π}{4}\)

The reference angle is

\(\color{red}{\frac{π}{4}}\)

16)For \(-\frac{25π}{6}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(4π=\frac{24π}{6}\) to get \(-\frac{π}{6}\), then add \(2π\) to get \(\frac{11π}{6}\).

One positive coterminal angle is

\(\frac{11π}{6}\)

, and one negative coterminal angle is

\(-\frac{π}{6}\)

The reference angle is

\(\color{red}{\frac{π}{6}}\)

17)For \(1120°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(1080°\): \(1120°-1080°=40°\).

One positive coterminal angle is

\(40°\)

, and one negative coterminal angle is

\(-320°\)

The reference angle is

\(\color{red}{40°}\)

18)For \(-940°\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(1080°\): \(-940°+1080°=140°\), which is in Quadrant II.

One positive coterminal angle is

\(140°\)

, and one negative coterminal angle is

\(-220°\)

The reference angle is

\(\color{red}{40°}\)

19)For \(\frac{29π}{3}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Subtract \(8π=\frac{24π}{3}\): \(\frac{29π}{3}-\frac{24π}{3}=\frac{5π}{3}\).

One positive coterminal angle is

\(\frac{5π}{3}\)

, and one negative coterminal angle is

\(-\frac{π}{3}\)

The reference angle is

\(\color{red}{\frac{π}{3}}\)

20)For \(-\frac{37π}{4}\), find one positive coterminal angle, one negative coterminal angle, and the reference angle.

Add \(10π=\frac{40π}{4}\): \(-\frac{37π}{4}+\frac{40π}{4}=\frac{3π}{4}\).

One positive coterminal angle is

\(\frac{3π}{4}\)

, and one negative coterminal angle is

\(-\frac{5π}{4}\)

The reference angle is

\(\color{red}{\frac{π}{4}}\)

How to Find Co-terminal Angles and Reference Angles