Coterminal Angles - Formula | How to Find Coterminal Angles? (2023)

The coterminal angles are the angles that have the same initial side and the same terminal sides. We determine the coterminal angle of a given angle by adding or subtracting 360° or 2π to it. In trigonometry, the coterminal angles have the same values for the functions of sin, cos, and tan.

Once you have understood the concept, you will differentiate between coterminal angles and reference angles, as well as be able to solve problems with the coterminal angles formula.

1.What Are Coterminal Angles?
2.Coterminal Angles Formula
3.How to Find Coterminal Angles?
4.Positive and Negative Coterminal Angles
5.Coterminal Angles and Reference Angles
6.FAQs on Coterminal Angles

What Are Coterminal Angles?

Coterminal angles are the angles that have the same initial side and share the terminal sides. These angles occupy the standard position, though their values are different. They are on the same sides, in the same quadrant and their vertices are identical. When the angles are moved clockwise or anticlockwise the terminal sides coincide at the same angle. An angle is a measure of the rotation of a ray about its initial point. The original ray is called the initial side and the final position of the ray after its rotation is called the terminal side of that angle.

Consider 45°. Its standard position is in the first quadrant because its terminal side is also present in the first quadrant. Look at the image.

  • On full rotation anticlockwise, 45° reaches its terminal side again at 405°. 405° coincides with 45° in the first quadrant.
  • On full rotation clockwise, 45° reaches its terminal side again at -315°. -315° coincides with 45° in the first quadrant.

Thus 405° and -315° are coterminal angles of 45°.

Coterminal Angles - Formula | How to Find Coterminal Angles? (1)

Coterminal Angles Formula

The formula to find the coterminal angles of an angle θ depending upon whether it is in terms of degrees or radians is:

  • Degrees: θ ± 360 n
  • Radians: θ ± 2πn

In the above formula, θ ± 360n, 360n means a multiple of 360, where n is an integer and it denotes the number of rotations around the coordinate plane.

Thus we can conclude that 45°, -315°, 405°, - 675°, 765° ..... are all coterminal angles. They differ only by a number of complete circles. We can conclude that "two angles are said to be coterminal if the difference between the angles is a multiple of 360° (or 2π, if the angle is in terms of radians)". Let us learn the concept with the help of the given example.

(Video) Coterminal Angles - Positive and Negative, Converting Degrees to Radians, Unit Circle, Trigonometry

Example : Find two coterminal angles of 30°.

Solution:

The given angle is, θ = 30°

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles.

For finding one coterminal angle: n = 1 (anticlockwise)

Then the corresponding coterminal angle is,

= θ + 360n
= 30 + 360 (1)
= 390°

Finding another coterminal angle :n = −2 (clockwise)

Then the corresponding coterminal angle is,

= θ + 360n
= 30 + 360(−2)
= −690°

(Video) Coterminal Angles In Radians & Degrees - Basic Introduction, Trigonometry

How to Find Coterminal Angles?

From the above explanation, for finding the coterminal angles:

  • add or subtract multiples of 360° from the given angle if the angle is in degrees.
  • add or subtract multiples of 2π from the given angle if the angle is in radians.

So we actually do not need to use the coterminal angles formula to find the coterminal angles. Instead, we can either add or subtract multiples of 360° (or 2π) from the given angle to find its coterminal angles. Let us understand the concept with the help of the given example.

Example: Find a coterminal angle of π/4.

Solution:

The given angle is θ = π/4, which is in radians.

So we add or subtract multiples of 2π from it to find its coterminal angles.

Let us subtract 2π from the given angle.

π/4 − 2π = −7π/4

Thus, a coterminal angle of π/4 is −7π/4.

(Video) Understanding coterminal angles in trigonometry

Positive and Negative Coterminal Angles

The coterminal angles can be positive or negative. In one of the above examples, we found that 390° and -690° are the coterminal angles of 30°

Here,

  • 390° is the positive coterminal angle of 30° and
  • -690° is the negative coterminal angle of 30°

θ ± 360 n, where n takes a positive value when the rotation is anticlockwise and takes a negative value when the rotation is clockwise. So we decide whether to add or subtract multiples of 360° (or 2π) to get positive or negative coterminal angles.

Coterminal Angles and Reference Angles

We already know how to find the coterminal angles of a given angle. The reference angle of any angle always lies between 0° and 90°, It is the angle between the terminal side of the angle and the x-axis. The reference angle depends on the quadrant's terminal side.

The steps to find the reference angle of an angle depends on the quadrant of the terminal side:

  • We first determine its coterminal angle which lies between 0° and 360°.
  • We then see the quadrant of the coterminal angle.
  • If the terminal side is in the first quadrant ( 0° to 90°), then the reference angle is the same as our given angle. For example, if the given angle is 25°, then its reference angle is also 25°.
  • If the terminal side is in the second quadrant ( 90° to 180°), then the reference angle is (180° - given angle). For example, if the given angle is 100°, then its reference angle is 180° – 100° = 80°.
  • If the terminal side is in the third quadrant (180° to 270°), then the reference angle is (given angle - 180°). For example, if the given angle is 215°, then its reference angle is 215° – 180° = 35°.
  • If the terminal side is in the fourth quadrant (270° to 360°), then the reference angle is (360° - given angle). For example, if the given angle is 330°, then its reference angle is 360° – 330° = 30°.

Example: Find the reference angle of 495°.

Solution:

Let us find the coterminal angle of 495°. The coterminal angle is 495° − 360° = 135°.

The terminal side lies in the second quadrant. Thus the reference angle is 180° -135° = 45°

Therefore, the reference angle of 495° is 45°.

(Video) Coterminal Angles - Example 1

Important Notes on Coterminal Angles:

  • The difference (in any order) of any two coterminal angles is a multiple of 360°
  • To find the coterminal angle of an angle, we just add or subtract multiples of 360°. from the given angle.
  • The number of coterminal angles of an angle is infinite because there is an infinite number of multiples of 360°.
  • If two angles are coterminal, then their sines, cosines, and tangents are also equal.

Related Articles:

  • Types of Angles and their Properties
  • Pairs of Angles
  • Corresponding Angles
  • Coterminal Angles Calculator

FAQs on Coterminal Angles

What is the Process of Finding Coterminal Angles?

For finding coterminal angles, we add or subtract multiples of 360° or 2π from the given angle according to whether it is in degrees or radians respectively. For example, some coterminal angles of 10° can be 370°, -350°, 730°, -710°, etc. In other words, the difference between an angle and its coterminal angle is always a multiple of 360°.

What is the Coterminal Angle of 45°?

Let us find a coterminal angle of 45° by adding 360° to it. 45° + 360° = 405°. Thus, 405° is a coterminal angle of 45°.

What is the Coterminal Angle of 60°?

Let us find a coterminal angle of 60° by subtracting 360° from it. 60° − 360° = −300°. Thus, -300° is a coterminal angle of 60°.

What is the Coterminal Angle of -30° Between 0° and 360°?

To find a coterminal angle of -30°, we can add 360° to it. − 30° + 360° = 330°. Thus, 330° is the required coterminal angle of -30°.

How do you Find Positive Coterminal Angles?

To find positive coterminal angles we need to add multiples of 360° to a given angle. For example, the positive coterminal angle of 100° is 100° + 360° = 460°.

How do you Find Negative Coterminal Angles?

To find negative coterminal angles we need to subtract multiples of 360° from a given angle. For example, the negative coterminal angle of 100° is 100° - 360° = -260°.

FAQs

What is the formula for finding Coterminal angles? ›

We can find the coterminal angles of a given angle by using the following formula: Coterminal angles of a given angle θ may be obtained by either adding or subtracting a multiple of 360° or 2π radians. Coterminal of θ = θ + 360° × k if θ is given in degrees. Coterminal of θ = θ + 2π × k if θ is given in radians.

How do you find the positive and negative Coterminal angles? ›

To find a positive and a negative angle coterminal with a given angle, you can add and subtract 360°if the angle is measured in degrees or 2π if the angle is measured in radians. Example 1: Find a positive and a negative angle coterminal with a 55°angle. A −305°angle and a 415°angle are coterminal with a 55°angle.

Do you multiply to find Coterminal angles? ›

For finding coterminal angles, we add or subtract multiples of 360° or 2π from the given angle according to whether it is in degrees or radians respectively.

What are 5 examples of Coterminal angles? ›

In the above figure, 45°, 405° and -315° are coterminal angles having the same initial side (x-axis) and the same terminal side but with different amount of rotations. Other Examples: Similarly, 30°, -330°, 390° and 57°, 417°, -303° are also coterminal angles.

How to find a positive angle less than 2pi that is coterminal with the given angle? ›

In order to find a positive angle that is coterminal with the given angle which is less than 2π , we subtract 2π and apply two full rotations (n=2) to the given angle. Hence, a positive angle less than 2π that is coterminal with the given angle is 4π3 4 π 3 .

Are two Coterminal angles the same rotation? ›

So coterminal angles are two angles that have the same initial side and the same terminal side, but different amounts of rotation as noted by Emory University.

How many angles are Coterminal to if 45 degrees? ›

In Mathematics, the coterminal angle is defined as an angle, where two angles are drawn in the standard position. Also both have their terminal sides in the same location. For example, the coterminal angle of 45 is 405 and -315. Here 405 is the positive coterminal angle, -315 is the negative coterminal angle.

What is the Coterminal angle of 420? ›

Substituting these angles into the coterminal angles formula gives 420 ° = 60 ° + 360 ° × 1 420\degree = 60\degree + 360\degree\times 1 420°=60°+360°×1.

What is the Coterminal of 180 degrees? ›

The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa. For an angle of 390° we find its positive coterminal angle by subtracting 360°. This gives us the coterminal angle of 30°.

What angle between 0 and 360 is Coterminal with 600? ›

Summary: The angle between 0 degrees and 360 degrees that is coterminal with -600 degrees -240 degrees and 120 degrees b.

What angle between 0 and 360 is coterminal to the given angle − 110? ›

Summary: The measure of an angle between 0 degrees and 360 degrees co terminal with an angle of -110 in standard position is 250°.

How do you find an angle between 0 and 360 that is Coterminal with 420? ›

Trigonometry Examples

Find an angle that is positive, less than 360° , and coterminal with 420° . Subtract 360° 360 ° from 420° 420 ° . The resulting angle of 60° 60 ° is positive, less than 360° 360 ° , and coterminal with 420° 420 ° .

What is a Coterminal angle in math? ›

Coterminal angles: are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side. For example, the angles 30°, –330° and 390° are all coterminal (see figure 2.1 below).

What is the Coterminal angle of 330? ›

Add 360° 360 ° to −330° - 330 ° . The resulting angle of 30° 30 ° is positive and coterminal with −330° - 330 ° . Since 30° is in the first quadrant, the reference angle is 30° .

What is the Coterminal angle of 550? ›

Subtract 360° 360 ° from 550° 550 ° . The resulting angle of 190° 190 ° is positive, less than 360° 360 ° , and coterminal with 550° 550 ° .

What is a positive angle less than 2π that is coterminal with 27π 4? ›

465∘, 27π/4. Summary: A positive angle less than 2π that is coterminal with: 465° is 105° and 27π/4 is 3/4π.

What is the Coterminal angle of 11pi 3? ›

Therefore, a coterminal angle for 11π3 11 π 3 is 5π3 5 π 3 .

Are Coterminal angles equal? ›

If the angles are the same, say both 60°, they are obviously coterminal. But the angles can have different measures and still be coterminal. In the figure above, rotate A around counterclockwise past 360° until it lies on top of DB.

What is the Coterminal angle of 450? ›

Therefore the positive and negative coterminal angles of 450∘ are 90∘ and −90∘ , respectively.

What angle is Coterminal to 145 degrees? ›

The angles which will be coterminal with 145° angle will be -215°, 505°, -575°, 865°, -1135° ……. The angles coterminal with 145° will be an infinite list. We add or subtract 360° to the given angle to get the coterminal angles. We add or subtract multiples of 360° to find all the coterminal angles.

What is the Coterminal angle of 760? ›

Answer and Explanation:

t = 760 ∘ . The angle is greater than two times 360∘ . So, we have to subtract 720∘ from the given angle t to obtain the coterminal angle θ . Hence, the coterminal angle is 40∘ .

What is the Coterminal angle of 595? ›

To find the coterminal angles we can add/subtract 360∘. In this case, our angle is greater than 360∘ so it makes sense to subtract 360∘ to get a positive coterminal angle: 595∘−360∘=235∘.

What is the Coterminal angle of 135? ›

Trigonometry Examples

Find an angle that is positive, less than 360° , and coterminal with −135° . Add 360° 360 ° to −135° - 135 ° . The resulting angle of 225° 225 ° is positive and coterminal with −135° - 135 ° .

What is the Coterminal angle of 410? ›

Subtract 360° 360 ° from 410° 410 ° . The resulting angle of 50° 50 ° is positive, less than 360° 360 ° , and coterminal with 410° 410 ° .

What angle is Coterminal to 510 degrees? ›

Subtract 360° 360 ° from 510° 510 ° . The resulting angle of 150° 150 ° is positive, less than 360° 360 ° , and coterminal with 510° 510 ° .

What is the angle between 0 and 360 that is Coterminal with 375? ›

To find coterminal angles, we have to either add or subtract 360∘ . However, adding 360∘ would result to a value outside the domain that we are considering. So, we simply subtract 360∘ from 375∘ . Therefore, the required angle is 15∘ .

What is the Coterminal of 400? ›

The negative coterminal angle of 400∘ is −320∘

What is the Coterminal angle of 460? ›

Coterminal angles can be found either by subtracting or adding 360∘ to the given angle. So 460∘ is a positive coterminal angle of 820∘ .

How do you find the positive and negative Coterminal angle of 780? ›

Trigonometry Examples

Find an angle that is positive, less than 360° , and coterminal with 780° . Subtract 360° 360 ° from 780° 780 ° . The resulting angle of 420° 420 ° is positive and coterminal with 780° 780 ° but isn't less than 360° 360 ° . Repeat the step.

What angle between 0 and 360 is coterminal with 990? ›

Since the angle is coterminal with 990", the angle is 90°.

What is the Coterminal angle of 11pi 4? ›

The angle 11pi/4, coterminal to angle 3pi/4, is located in the Second Quadrant(Quadrant II). Since cos function is negative in the 2nd quadrant, thus cos 11pi/4 value = −(1/√2) or -0.7071067. . . Similarly, cos 11pi/4 can also be written as, cos 11pi/4 = (11pi/4 + n × 2pi), n ∈ Z.

What angle between 0 and 360 is coterminal with 440? ›

The resulting angle of 80° 80 ° is positive, less than 360° 360 ° , and coterminal with 440° 440 ° .

What angle is Coterminal with 23pi 4? ›

The resulting angle of 7π4 7 π 4 is positive, less than 2π 2 π , and coterminal with 23π4 23 π 4 .

What angle is Coterminal to 110? ›

Answer and Explanation: We are given an angle that is -110 degrees. To find the coterminal angle of -110 degrees that is between 0 and 360 degrees, we could add 360 degrees to -110. The coterminal angle of -110 degrees that is in between 0 degree and 360 degrees is 250 degrees.

What is the Coterminal angle of 170? ›

Add 360° 360 ° to −170° - 170 ° . The resulting angle of 190° 190 ° is positive and coterminal with −170° - 170 ° .

What is the Coterminal angle of 800? ›

An angle of 800 degrees is coterminal with an angle of 800- 360 = 440 degrees. It would also be coterminal with an angle of 440-360 = 80 degrees.

How do you find the Coterminal angle of 240? ›

Trigonometry Examples

Find an angle that is positive, less than 360° , and coterminal with −240° . Add 360° 360 ° to −240° - 240 ° . The resulting angle of 120° 120 ° is positive and coterminal with −240° - 240 ° .

What is the Coterminal of 115? ›

The coterminal angles of 115∘ are 475∘ and −245∘ .

What is the Coterminal angle of 125? ›

Add 360° 360 ° to −125° - 125 ° . The resulting angle of 235° 235 ° is positive and coterminal with −125° - 125 ° .

How do you find a Coterminal angle of 800 degrees? ›

Since adding or subtracting a full rotation, 360 degrees, would result in an angle with terminal side pointing in the same direction, we can find coterminal angles by adding or subtracting 360 degrees. An angle of 800 degrees is coterminal with an angle of 800- 360 = 440 degrees.

What is the Coterminal of 45? ›

In Mathematics, the coterminal angle is defined as an angle, where two angles are drawn in the standard position. Also both have their terminal sides in the same location. For example, the coterminal angle of 45 is 405 and -315.

What are the Coterminal angles of 760? ›

t = 760 ∘ . The angle is greater than two times 360∘ . So, we have to subtract 720∘ from the given angle t to obtain the coterminal angle θ . Hence, the coterminal angle is 40∘ .

Which angle is Coterminal to 128? ›

All angles having a measure of 128° + 360k°, where k is an integer, are coterminal with 128°. A positive angle is 28° + 360°(3) or 1208°.

What is the Coterminal angle of 720? ›

Subtract 360° 360 ° from 720° 720 ° . The resulting angle of 360° 360 ° is positive, less than 360° 360 ° , and coterminal with 720° 720 ° .

What is the Coterminal angle of 450 degrees? ›

Answer and Explanation: Therefore the positive and negative coterminal angles of 450∘ are 90∘ and −90∘ , respectively.

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