Coterminal Angle Calculator

Coterminal angles share the same terminal side (differ by full turns).

How the Math Works

Coterminal angles are angles in standard position that share the same terminal side but have different measures. To find coterminal angles, you simply add or subtract multiples of 360 degrees (or 2π radians) from the given angle. For example, a 30-degree angle is coterminal with 390 degrees (30 + 360) and -330 degrees (30 - 360). This works because rotating a full circle (360 degrees) brings you back to the same position, so adding or subtracting full rotations doesn't change where the angle points.

Practical Applications

To use this calculation, start with any angle measure and apply the formula: new angle = original angle ± 360n, where n is any integer. For positive coterminal angles, add 360 repeatedly until you get a positive result. For negative coterminal angles, subtract 360 repeatedly. This is particularly useful when working with trigonometric functions, simplifying angle measures in calculations, or converting between different angle representations in mathematics and engineering problems.

Day-to-Day Use

Understanding coterminal angles helps in everyday situations involving rotations and directions. When setting up a tent, organizing items around a circular table, or adjusting a clock's hands, you're essentially working with angles and their relationships. Artists and designers use these concepts when creating circular patterns or arranging elements in circular compositions. Even in sports like golf or baseball, knowing how angles relate to each other through full rotations can improve your understanding of trajectories and positioning.

Worked example

45° → 405° and −315°.

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