Coterminal Angle
Find coterminal angles in degrees or radians. Get the smallest positive, largest negative, and first coterminal angles instantly with our free calculator.
About This Calculator
The Coterminal Angle Calculator is a free online tool that helps you find angles sharing the same terminal side as any given angle. Whether you are a student learning trigonometry, a teacher preparing math lessons, or a professional working with periodic functions and angle measurements, this calculator provides instant results for both degrees and radians.
Two angles are coterminal when they share the same initial and terminal sides in standard position. The calculator computes the smallest positive coterminal angle by taking the input modulo 360\u00b0 (or 2\u03c0 radians), ensuring the result falls within the 0\u00b0 to 360\u00b0 range. It subtracts 360\u00b0 to find the largest negative coterminal angle and adds or subtracts one full rotation to find the first positive and negative coterminal angles. The general formula used is \u03b8 \u00b1 360\u00b0n (degrees) or \u03b8 \u00b1 2\u03c0n (radians), where n is any integer.
Coterminal angles are fundamental in trigonometry because they share the same sine, cosine, and tangent values, making them essential for solving trigonometric equations, analyzing periodic wave functions, and understanding the unit circle. This calculator supports any real number input, including negative angles, angles greater than 360\u00b0, and fractional values, with the ability to toggle between degrees and radians.
Frequently Asked Questions
What is a coterminal angle?
A coterminal angle is an angle that shares the same terminal side as a given angle when drawn in standard position. Two angles are coterminal if they differ by a multiple of 360 deg (or 2pi radians). For example, 45 deg and 405 deg are coterminal because 405 deg = 45 deg + 360 deg.
How do you find a coterminal angle?
To find a coterminal angle, add or subtract multiples of 360 deg (in degrees) or 2pi (in radians) from the given angle. The general formula is theta +/- 360 deg x n, where n is any integer. A positive coterminal angle is found by adding 360 deg, and a negative coterminal angle by subtracting 360 deg.
What is the smallest positive coterminal angle?
The smallest positive coterminal angle is obtained by taking the given angle modulo 360 deg (wrapping negative values into the 0 deg to 360 deg range). For example, for 724 deg, the smallest positive coterminal angle is 4 deg. For -45 deg, it is 315 deg.
How do you find coterminal angles in radians?
To find coterminal angles in radians, add or subtract multiples of 2pi radians. For an angle theta in radians, coterminal angles are theta +/- 2pin, where n is any integer. For example, pi/4 and 9pi/4 are coterminal because 9pi/4 = pi/4 + 2pi.
How many coterminal angles does an angle have?
Every angle has an infinite number of coterminal angles because you can add or subtract 360 deg (or 2pi radians) any number of times. The set of all coterminal angles for a given angle theta is theta +/- 360 degn (or theta +/- 2pin) where n is any integer.
What are coterminal angles used for?
Coterminal angles are used extensively in trigonometry, calculus, physics, and engineering to simplify angle measurements, analyze periodic functions, solve trigonometric equations, and convert between degree and radian measurements. They are particularly useful when working with angles greater than 360 deg or negative angles.
Are 30 deg and 390 deg coterminal?
Yes, 30 deg and 390 deg are coterminal because 390 deg = 30 deg + 360 deg. They share the same terminal side when drawn in standard position and have the same sine, cosine, and tangent values.
How does this coterminal angle calculator work?
This calculator takes your input angle in degrees or radians, computes the smallest positive coterminal angle by taking the modulo 360 deg, then derives the largest negative (by subtracting 360 deg) and the first positive/negative coterminal angles (by adding/subtracting 360 deg). For radian inputs, it converts to degrees internally and converts results back to radians.