# Online Coterminal Angle Calculator

## Coterminal Angle Calculator

An online coterminal angle calculator is a handy tool that makes dealing with angles easy. You just type in the angle value, either in degrees or radians, and the calculator shows you all the coterminal angles for that angle. It even goes a step further by giving you the negative coterminal angles, which are found by subtracting 360 degrees from the given angle.

What’s cool is that these online calculators also let you switch between degrees and radians, making it super convenient for your calculations. So, whether you’re working with angles in degrees or radians, the online coterminal angle calculator has got you covered.

## What is a Coterminal Angle?

Coterminal angles are like friends that start and end at the same place in a circle. Imagine drawing an angle starting from the positive x-axis – that’s where it begins, and the journey it takes around the circle brings it back to the same ending point.

**Initial Side**

This is where the angle begins, sort of like its starting point on the circle.

**Terminal Side**

The terminal side is where the angle finishes its trip around the circle, reaching the same ending point.

So, coterminal angles have the same starting and ending points on the circle, like 110° and -250°. They’re like different paths that bring you to the same destination.

## Positive and Negative Coterminal Angle

Positive and negative coterminal angles are like taking a stroll around a circle but in different directions.

**Positive Coterminal Angle**

Positive coterminal angles refer to angles that share the same terminal side when measured in standard position but have different initial sides. In other words, positive coterminal angles are obtained by adding a certain number of complete revolutions (360 degrees or 2π radians) to the original angle.

For example, if you have an angle of 30 degrees, the positive coterminal angles can be found by adding 360 degrees or any multiple of 360 degrees. So, positive coterminal angles for 30 degrees would include 390 degrees, 750 degrees, and so on.

Mathematically, if α is an angle in standard position, then a positive coterminal angle β can be expressed as

Let’s say you have an angle, α, like 14°.

Now, if you go around the circle once (which is 360°), starting from your angle, you end up at a new angle, β.

So, for a positive coterminal angle, you add 360° to your starting angle: β = α + 360°.

In our example: *β*=14°+360°=374°.

**Negative Coterminal Angle**

A negative coterminal angle is an angle that shares the same terminal side with another angle in the standard position but is obtained by subtracting a certain number of complete revolutions (360 degrees or 2π radians) from the original angle. In other words, negative coterminal angles have different initial sides but end at the same location on the coordinate plane.

For example, if you have an angle of 100 degrees, the negative coterminal angles can be found by subtracting 360 degrees or any multiple of 360 degrees. So, negative coterminal angles for 100 degrees would include -260 degrees, -620 degrees, and so forth.

Now, what if you go around the circle in the opposite direction? That’s like subtracting 360° from your starting angle.

For a negative coterminal angle: β = α – 360°.

In our example: *β*=14°−360°=−346°.

So, by adding or subtracting full spins (360°), you find positive and negative coterminal angles. It’s like taking different routes but always ending up at the same place on the circle.

## Coterminal Angle Formula

The coterminal angle formula allows you to find angles that share the same terminal side in a standard position. The formula is based on adding or subtracting complete revolutions (360 degrees or 2π radians) to the original angle. Here’s the coterminal angle formula

**In Degrees**

For coterminal angles in degrees, you can use this formula: *β*=*α*±(360°×*k*), where *k* is any whole number.

This means you’re adding or subtracting complete spins (360°) to or from the given angle.

**In Radians**

In the case of radians, the formula is *β*=*α*±(2*π*×*k*), where *k* is any whole number.

Here, it’s like adding or subtracting a full circle (2π) to or from the angle.

**Key Points**

The starting point is where the angle begins (the initial side).

After spinning around, you end up at the coterminal angle.

Coterminal angles share the same trigonometric values, making them like math twins.

So, if you want to find coterminal angles, just add or subtract a full spin (or more) using this cool formula. It’s like finding different routes but always reaching the same spot on the spinning wheel.

## How to Find Coterminal Angles?

Finding coterminal angles is like exploring different paths around a circle. Here’s how you can do it.

**Starting Angle (α)**

Begin with the angle you have, let’s call it α. For example, if α is 50 degrees.

**Add or Subtract 360 Degrees (In Degrees)**

To find coterminal angles, you can add or subtract any multiple of 360 degrees (a full circle) to the starting angle.

*α*±360*n* degrees, where *n* is an integer (a whole number).

**In Radians (In Case of Radians)**

If you’re working in radians, you can use a similar approach.

*α*±2*πn* radians, where *n* is an integer.

**Example**

Let’s take the example of an angle of 50 degrees (α = 50°).

Adding 360 degrees, we get 50°+360°=410°

Subtracting 360 degrees, we get 50°−360°=−310°.

Both 410 degrees and -310 degrees are coterminal angles of the original 50-degree angle.

So, by adding or subtracting full spins, you can find different coterminal angles. It’s like taking different routes around the circle, but you always end up at the same place.

**How the Coterminal Angle Calculator Works**

Our Coterminal Angle Calculator is like a math wizard that helps you with angles. Here’s how to use our Coterminal Angle Calculator.

**Input**

**Choose Calculation Type**

Decide what you want to calculate: find coterminal angles or check if two angles are coterminal.

**Select Degree or Radian**

Pick whether your angle values are in degrees or radians.

**Enter Angle Value**

Put in the value of your angle in the designated field.

**Tap “Calculate”**

Hit the magic button to let the calculator do its thing.

**Output**–

Our Coterminal Angle Calculator will give you the following results.

**Positive and Negative Coterminal Angle**

It tells you the angle you entered and its positive and negative coterminal angles.

**Check Coterminality**

It lets you know if two angles are coterminal or not.

So, just feed in your angle values, choose what you want to find, tap “Calculate,” and let the calculator work its magic! It’s like having a math genie in a digital lamp.

### FAQ

**What are coterminal angles?**

Coterminal angles are angles that share the same terminal side in standard position but have different initial sides. They are obtained by adding or subtracting complete revolutions (360 degrees or 2π radians) from a given angle.

**What is a Coterminal Angle Calculator?**

A Coterminal Angle Calculator is an online tool designed to calculate angles that share the same terminal side in standard position, making trigonometric calculations and angle conversions more accessible.

**How does the Coterminal Angle Calculator work?**

The Coterminal Angle calculator operates by allowing users to input an angle in degrees or radians. It then calculates positive and negative coterminal angles, providing results that share the same terminal side.

**Can coterminal angles be negative?**

Yes, coterminal angles can be negative. Negative coterminal angles are obtained by subtracting complete revolutions from the original angle.

**What is the significance of coterminal angles?**

Coterminal angles are important in trigonometry as they have the same trigonometric values. They help simplify calculations and understand periodic phenomena.

**How do you find a coterminal angle for an angle in radians?**

The coterminal angle is equal to the angle *θ* in radians plus or minus 2 times π times and integer *n* representing the revolutions about the point.

**Are there multiple coterminal angles for a given angle?**

Yes, there are infinitely many coterminal angles for a given angle. You can find them by adding or subtracting any multiple of 360 degrees or 2π radians.

**How do coterminal angles relate to the unit circle?**

Coterminal angles on the unit circle end at the same point, indicating the same position in the coordinate plane. They contribute to understanding periodic behavior and rotational symmetry.

**Can coterminal angles be used in real-world applications?**

Yes, coterminal angles are used in various fields such as physics, engineering, and computer graphics to represent periodic phenomena and rotational movements.

**Which angles are Coterminal with a 125-degree angle?**

The angles that are coterminal with 125° are 485° and -235°.

**Is there a difference between positive and negative coterminal angles?**

Positive coterminal angles are obtained by adding complete revolutions, while negative coterminal angles are obtained by subtracting complete revolutions. Both have the same terminal side.

**How can I check if two angles are coterminal?**

Use the coterminal angle formula and check if the resulting angles share the same terminal side. If they do, the angles are coterminal.

**What is the least positive coterminal angle?**

The least positive coterminal angle is the smallest positive angle that shares the same terminal side as the given angle in standard position, achieved by subtracting complete revolutions (360 degrees or 2π radians) until reaching the smallest positive value.

**Can coterminal angles be used in navigation?**

Coterminal angles are not directly used in navigation, but understanding angles is essential for navigation and map reading.