Coterminal Angle Calculator – Find Positive & Negative Coterminal Angles

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How to Calculate Coterminal Angles & Use the Calculator (With Examples)

Coterminal Angle Calculator – Free Online Tool with Formula, Steps & Examples

Coterminal Angle Calculator

What is Coterminal Angle?

A coterminal angle refers to any two or more angles that share the same initial and terminal sides when drawn in standard position, differing only by a full rotation (360° in degrees or 2π radians). For instance, 30°, 390°, and -330° are all coterminal because they all end up at the exact same terminal side if plotted from the x-axis on a coordinate plane. In other words, coterminal angles "land" at the same position on the unit circle, even though their measures are different.

Formula or Logic Behind Coterminal Angle Calculator

The formula for finding coterminal angles is straightforward:
In degrees: Coterminal Angle = θ ± 360° × n
In radians: Coterminal Angle = θ ± 2π × n
Here, θ is the given angle and n is any integer (positive, negative, or zero). To find the smallest positive coterminal angle, you can use the modulo operation:
Degrees: ((θ mod 360) + 360) mod 360
Radians: ((θ mod 2π) + 2π) mod 2π
This ensures the coterminal angle is always within [0°, 360°) or [0, 2π), suitable for most trigonometric applications.

Common Angles & Their Coterminal Angles

Angle (Degrees)	Coterminal Angles
30°	390°, -330°, 750°
45°	405°, -315°, 765°
120°	480°, -240°, 840°
270°	630°, -90°, 990°
-45°	315°, 675°, -405°
Angle (Radians)	Coterminal Angles
π/6 ≈ 0.524	0.524, 2π+0.524 ≈ 6.808, -1.571
2π/3 ≈ 2.094	2.094, 2π+2.094 ≈ 8.377, -4.189
3π/2 ≈ 4.712	4.712, 2π+4.712 ≈ 10.996, -1.570

Steps to Use the Coterminal Angle Calculator

Enter your angle value in the input box.
Select the unit: Degrees (°) or Radians.
Click on the 'Calculate' button.
View the main coterminal angle, plus positive and negative coterminal angles instantly.

Why Use Vedantu’s Coterminal Angle Calculator?

Vedantu’s Coterminal Angle Calculator is designed for simplicity and accuracy. It's mobile-friendly, works perfectly on any device, and instantly displays coterminal angles for quick homework help, exam practice, or teaching demonstrations. Trusted by countless students and teachers across India, this tool uses classroom-approved logic and NCERT-based trigonometry, making it reliable for all academic and practical purposes.

Real-life Applications of Coterminal Angle Calculator

The coterminal angle calculator helps in many ways, such as simplifying trigonometric expressions, solving engineering or physics problems (like angular displacement and circular motion), and checking answers in competitive exams like JEE, NEET, and CBSE boards. It is also valuable in real-life situations involving rotations, periodic processes (waves, electricity, gears), and even in fields like astronomy.

For example, when working with the unit circle or analyzing angles in geometry, quickly finding coterminal values is essential. You can also use similar concepts in conjunction with the reference angle calculator or practice more trigonometric concepts here: Trigonometry.

Explore more math tools and resources on Vedantu, such as the HCF Calculator, Prime Numbers, and Angle of Elevation for deeper learning.

FAQs on Coterminal Angle Calculator – Find Positive & Negative Coterminal Angles

1. What is a coterminal angle?

Coterminal angles are angles that share the same terminal side when drawn in standard position. They differ by multiples of 360 degrees (or 2π radians). In simpler terms, imagine rotating around a circle; coterminal angles end up at the same point after completing one or more full rotations.

2. How do you find coterminal angles?

To find coterminal angles, add or subtract multiples of 360 degrees (for degrees) or 2π radians (for radians) to the given angle. The formula is: Coterminal angle = θ + 360n° (degrees) or θ + 2πn (radians), where θ is the original angle and n is any integer.

3. What is the formula for finding coterminal angles?

The formula for finding coterminal angles is: θ + 360n° (for degrees) or θ + 2πn (for radians), where θ represents the original angle, and n is any integer (positive or negative). This allows you to find infinitely many coterminal angles.

4. How many coterminal angles does an angle have?

An angle has infinitely many coterminal angles. You can add or subtract multiples of 360° or 2π radians indefinitely to obtain different coterminal angles.

5. What is a positive coterminal angle?

A positive coterminal angle is an angle obtained by adding a positive multiple of 360° (or 2π radians) to the original angle. It represents a counterclockwise rotation from the positive x-axis.

6. What is a negative coterminal angle?

A negative coterminal angle is found by subtracting a positive multiple of 360° (or 2π radians) from the original angle. It indicates a clockwise rotation from the positive x-axis.

7. How to find the smallest positive coterminal angle?

To find the smallest positive coterminal angle, repeatedly subtract 360° (or 2π radians) from the given angle until you get a result between 0° and 360° (or 0 and 2π radians).

8. How to find the smallest negative coterminal angle?

To find the smallest negative coterminal angle, repeatedly add 360° (or 2π radians) to the given angle until you get a result between -360° and 0° (or -2π and 0 radians).

9. What is the coterminal angle of 45 degrees?

Some coterminal angles of 45 degrees are 405 degrees (45 + 360), 765 degrees (45 + 720), -315 degrees (45 - 360), and so on. There are infinitely many.

10. What are coterminal angles used for?

Coterminal angles are useful in simplifying trigonometric calculations and understanding periodic functions. They help in solving problems involving rotations and angles in various fields like physics and engineering.

11. Can you give an example of finding a coterminal angle in radians?

Let's find a coterminal angle for π/3 radians. Adding 2π (a full rotation) gives π/3 + 2π = 7π/3. Subtracting 2π gives π/3 - 2π = -5π/3. Both 7π/3 and -5π/3 are coterminal with π/3.

12. Explain the difference between coterminal and reference angles.

Coterminal angles share the same terminal side, differing by full rotations. Reference angles, however, are the acute angle formed by the terminal side and the x-axis. They are always positive and less than 90 degrees (or π/2 radians).