The coterminal angles calculator is a simple online web application for calculating positive and negative coterminal angles for a given angle.
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Angles in standard position with a same terminal side are called coterminal angles. Coterminal angles are two angles that have the identical initial and terminal sides.
Coterminal angles can be established by starting on the same side and rotating in a positive or negative direction for more than 360 degrees, finishing on the same side. If the angle rounds more than once, coterminal angles can contain multiples of 360 degrees.
Because coterminal angles represent integer multiples of 360°, or 2n, there are an infinite number of coterminal angles with any given angle.
Because coterminal angles represent integer multiples of 360° or 2π, there are an infinite number of coterminal angles with any given angle.
Coterminal Angles Calculator Use
- In the first field, type the angle value.
- Choose the angle’s unit from the dropdown menu.
- The calculator will display the angle’s negative and positive coterminal angles.
Coterminal Angles Formula
The formula for finding the coterminal angles of an angle is given below, depending on whether the angle is measured in degrees or radians.
For any angle θ, coterminal angles exist in radians with angles (2π ± θ), (4π ± θ), (6π ± θ) and so on, or in degrees, ((1)360° ± θ), ((2)360° ± θ), and so on.
How to Find Coterminal Angles
Finding the coterminal of an angle is a simple task. Using the formula above, you can quickly find the positive and negative coterminal angles of any specified angle. With the help of some easy examples, we will show you how to perform it manually.
Example
Find four positive and negative angles that are coterminal with 50° and -25° and verify the answers using the above calculator.
Solution :
We have to find the four positive and negative coterminal angles of 50° and -25°. We will use the above formula to find the coterminal angles.
Because the angles in the problem are in degrees, we’ll apply the degrees formula.
Degrees = n360°± θ
Positive Coterminal Angles
- 50° + 360° = 410°
- 50° + (2 × 360°) = 770°
- 50° + (3 × 360°) = 1130°
- 50° + (4 × 360°) = 1490°
- -25° + 360° = 335°
- -25° + (2 × 360°) = 695°
- -25° + (3 × 360°) = 1055°
- -25° + (4 × 360°) = 1415°
Negative Coterminal Angles
- 50° – 360° = -310°
- 50° – (2 × 360°) = -670°
- 50° – (3 × 360°) = -1030°
- 50° – (4 × 360°) = -1390°
- -25° – 360° = -385°
- -25° – (2 × 360°) = -745°
- -25° – (3 × 360°) = -1105°
- -25° – (4 × 360°) = -1465°
Final Answer :
Positive Coterminal Angles of 55° = 410°, 770°, 1130°, 1490°.
Negative Coterminal Angles of 55° = -310°, -670°, -1030°, -1390°.
Positive Coterminal Angles of -25° = 335°, 695°, 1055°, 1415°.
Negative Coterminal Angles of -25° = -385°, -745°, -1105°, -1465°.
How to Calculate Coterminal Angles in Radians
The steps for calculating coterminal angles for angles in radians are the same as those for angles in degrees. For angles in radians, the specific formula is provided above. The formula can be used to calculate the angles. We’ve illustrated these steps with a basic example below.
Example
Find the two positive and negative angle coterminal with π/6 and verify the answers using the above calculator.
Solution :
We have to find the two positive and negative coterminal angles of π/6. We will use the above formula to find the coterminal angles.
Because the angles in the problem are in radians, we’ll apply the radians formula.
Radians = 2nπ± θ
Positive Coterminal Angles
- 2π + π/6 = 2π/1 + π/6 = (π + 12π)/6 = 13π/6 ≈ 6.8068
- 4π + π/6 = 4π/1 + π/6 = (π + 24π)/6 = 25π/6 ≈ 13.09
Negative Coterminal Angles
- 2π – π/6 = 2π/1 – π/6 = (π – 12π)/6 = -11π/6 ≈ -5.76
- 4π – π/6 = 4π/1 – π/6 = (π – 24π)/6 = -23π/6 ≈ -12.03
Final Answer :
Positive Coterminal Angles of π/6 = 6.806785307179585, 13.089970614359173.
Negative Coterminal Angles of π/6 = -5.759585307179586, -12.042770614359172.
Note : When the angle value is given in radians, the radians formula is recommended. However, some people may find it challenging to execute the fractions. To solve this problem, convert the value of radians to degrees and handle it as such. Once you’ve determined the coterminal angles, you can convert them to radians. To convert the values, use our degrees to radians and radians to degrees calculators.