Coterminal Angle Calculator

What is a Coterminal Angle?

In trigonometry, angles are often drawn in standard position on a coordinate plane, with the vertex at the origin and the initial side along the positive x-axis. The terminal side is where the angle measurement ends.

Coterminal angles are angles in standard position that share the same terminal side. Because you can get to the same terminal position by completing full rotations (either clockwise or counterclockwise), any given angle has infinitely many coterminal angles.

Think of it like spinning around: stopping after a 90° turn lands you in the same direction as stopping after a 450° turn (90° + 360°) or a -270° turn (90° - 360°). These angles (90°, 450°, -270°) are coterminal.

How This Coterminal Angle Calculator Works

This calculator quickly finds one positive and one negative coterminal angle for the angle you provide. Here's how:

1. Enter Angle: Input the angle value into the designated field. You can enter positive, negative, or zero values, including decimals.
2. Select Unit: Choose whether the angle you entered is in Degrees (°) or Radians (rad) using the radio buttons.
3. Click Calculate: Press the "Find Coterminal Angles" button.
4. Calculation:
   • If you selected Degrees, the calculator adds 360° to your angle to find a positive coterminal angle and subtracts 360° to find a negative coterminal angle. Positive = Angle + 360°
Negative = Angle - 360°
   • If you selected Radians, the calculator adds 2π radians (approximately 6.283) to your angle for the positive result and subtracts 2π radians for the negative result. Positive = Angle + 2π rad
Negative = Angle - 2π rad
5. View Results: The calculator displays the calculated positive and negative coterminal angles, including the correct unit (° or rad).

This tool finds the coterminal angles that are exactly one full rotation away from the original angle.

Frequently Asked Questions (FAQs)

Q1: How many coterminal angles does an angle have?

An angle has an infinite number of coterminal angles. You can keep adding or subtracting full rotations (360° or 2π radians) indefinitely to find more coterminal angles. This calculator finds the two closest ones by adding/subtracting just one rotation.

Q2: What's the difference between positive and negative coterminal angles?

It simply refers to the sign of the resulting angle value:

• Positive Coterminal Angle: An angle coterminal with the original angle that has a positive value. Often found by adding 360° or 2π until a positive result is achieved.
• Negative Coterminal Angle: An angle coterminal with the original angle that has a negative value. Often found by subtracting 360° or 2π until a negative result is achieved.

This calculator specifically adds 360° (or 2π) for the positive result and subtracts 360° (or 2π) for the negative result, regardless of the original angle's sign.

Q3: Can the angle I enter be negative or greater than 360° / 2π?

Yes, absolutely. You can input any real number for the angle, whether it's positive, negative, zero, a small angle, or an angle representing multiple rotations (e.g., 750° or -5π radians). The calculator will still find angles one rotation away.

Q4: How are radians handled? Does it show 'π'?

This calculator works with radians as decimal values. When you input a radian value (even if you think of it in terms of π, like 1.57 for π/2), the calculator adds or subtracts the decimal value of 2π (approximately 6.283185...). The results are displayed as decimal radians, not fractions of π.

Q5: Why are coterminal angles important?

Coterminal angles are important in trigonometry because trigonometric functions (sine, cosine, tangent, etc.) have the same value for coterminal angles. For example, sin(30°) is the same as sin(390°) and sin(-330°). This allows us to simplify problems by finding a convenient coterminal angle, often one between 0° and 360° (or 0 and 2π radians), known as the principal angle.

Q6: How do I find the *principal* coterminal angle (between 0° and 360°)?

This calculator finds angles one rotation away. To find the principal angle (the one between 0° and 360°, or 0 and 2π), you typically use the modulo operator or repeatedly add/subtract 360° (or 2π) until the angle falls within the desired range. For example:

• For 450°: 450° - 360° = 90°. The principal angle is 90°.
• For -30°: -30° + 360° = 330°. The principal angle is 330°.
• For 8π/3 rad: (8π/3) - 2π = (8π/3) - (6π/3) = 2π/3 rad. The principal angle is 2π/3.

A dedicated "Principal Angle Calculator" might use modulo arithmetic to find this directly.