Find One Positive and Negative Coterminal Angle Calculator

Coterminal angles are angles that share the same initial and terminal sides but differ by a full rotation (360° or 2π radians). This calculator helps you find one positive and one negative coterminal angle for any given angle.

What Are Coterminal Angles?

Coterminal angles are angles that have the same terminal side when drawn in standard position. They differ by integer multiples of 360° (for degrees) or 2π (for radians).

For example, 45° and 405° are coterminal because 405° - 360° = 45°. Similarly, -45° and 315° are coterminal because -45° + 360° = 315°.

Coterminal angles are useful in trigonometry, navigation, and engineering where periodic functions (like sine and cosine) repeat every full rotation.

How to Find Coterminal Angles

To find coterminal angles, you can add or subtract full rotations (360° or 2π) to the original angle. Here's how:

Start with your original angle (θ).
Add 360° to get a positive coterminal angle: θ + 360°.
Subtract 360° to get a negative coterminal angle: θ - 360°.

Formula for Coterminal Angles:

Positive coterminal angle = θ + 360°

Negative coterminal angle = θ - 360°

Example Calculation

Let's find coterminal angles for 45°:

Original angle: 45°
Positive coterminal angle: 45° + 360° = 405°
Negative coterminal angle: 45° - 360° = -315°

So, 405° and -315° are coterminal with 45°.

Common Mistakes

When working with coterminal angles, it's easy to make these mistakes:

Forgetting to add or subtract full rotations (360° or 2π).
Mixing up positive and negative coterminal angles.
Assuming all coterminal angles must be positive.

Always double-check your calculations by verifying that the difference between the original angle and the coterminal angle is a multiple of 360°.

Applications

Coterminal angles are used in various fields:

Trigonometry: Simplifying angle calculations.
Navigation: Determining directions and bearings.
Engineering: Designing rotating machinery.
Physics: Analyzing periodic motion.

FAQ

What is the difference between coterminal and supplementary angles?

Coterminal angles share the same terminal side and differ by full rotations. Supplementary angles add up to 180° and are not necessarily coterminal.

Can coterminal angles be negative?

Yes, coterminal angles can be negative. For example, -45° and 315° are coterminal.

How many coterminal angles can an angle have?

An angle has infinitely many coterminal angles, as you can keep adding or subtracting full rotations.

Are coterminal angles used in real-world applications?

Yes, they are used in navigation, engineering, and physics to simplify angle calculations and analyze periodic motion.