Coterminal Angle Calculator

What are Coterminal Angles?

In trigonometry, coterminal angles are angles in standard position (with their initial side along the positive x-axis and their vertex at the origin) that share the exact same terminal side.

Because one full revolution around the circle equals $360^\circ$ (or $2\pi$ radians), rotating by an extra full circle in either a clockwise or counterclockwise direction lands on the identical ray. Therefore, every angle has an infinite number of positive and negative coterminal angles.

Coterminal Angle Formulas

To find all coterminal angles of a given angle $\theta$, add or subtract integer multiples of a full turn:

In Degrees: $$\theta_{\text{coterminal}} = \theta + 360^\circ \times k \quad (\text{where } k \in \mathbb{Z})$$
In Radians: $$\theta_{\text{coterminal}} = \theta + 2\pi \times k \quad (\text{where } k \in \mathbb{Z})$$

Here, $k > 0$ yields larger positive coterminal angles, while $k < 0$ yields smaller and negative coterminal angles.

How to Find the Principal Coterminal Angle between 0° and 360°

When working with very large angles or negative angles, it is standard practice to find the equivalent angle within the standard interval $[0^\circ, 360^\circ)$ or $[0, 2\pi)$:

For positive angles larger than 360°: Subtract $360^\circ$ repeatedly, or compute the modulo remainder: $\theta \pmod{360^\circ}$. For example, $850^\circ - 2 \times 360^\circ = 850^\circ - 720^\circ = 130^\circ$.
For negative angles: Add $360^\circ$ until the value lies within the range $[0^\circ, 360^\circ)$. For example, $-120^\circ + 360^\circ = 240^\circ$.

How to Check If Two Angles Are Coterminal

Two angles $\alpha$ and $\beta$ are coterminal if and only if their difference is an exact multiple of $360^\circ$ (or $2\pi$ rad):

$$\frac{|\alpha - \beta|}{360^\circ} = k \in \mathbb{Z}$$

If the quotient is an integer, the angles share the same terminal ray; if the result contains a fractional decimal, the angles are not coterminal.

Related Trigonometry Tools

Explore related mathematical calculators on OnlineMiniTools:

Cotangent Calculator: Calculate $\cot(\theta)$ and trig values for any angle.
Tan Calculator: Find the tangent of angles in degrees and radians.
Cos Calculator: Calculate cosine values and solve triangle ratios.

Frequently Asked Questions

What is the coterminal angle of 45 degrees?

The coterminal angles of $45^\circ$ follow the formula $45^\circ + 360^\circ \times k$. Common examples include $405^\circ$ ($k=1$), $765^\circ$ ($k=2$), $-315^\circ$ ($k=-1$), and $-675^\circ$ ($k=-2$).

What is the difference between a coterminal angle and a reference angle?

A coterminal angle is any angle that shares the exact same terminal side as the original angle. A reference angle is always a positive acute angle ($0^\circ \le \theta_{\text{ref}} \le 90^\circ$) formed between the terminal side and the horizontal x-axis.

Do coterminal angles have the same trigonometric values?

Yes. Since all coterminal angles share the exact same terminal side and coordinates on the unit circle, $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$, and $\cot(\theta)$ are identical for all coterminal angles.

How many coterminal angles does a given angle have?

Every angle has an infinite number of coterminal angles because you can add or subtract full $360^\circ$ rotations infinitely in either direction.

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