Coterminal Angles

Trigonometry · Axiom Academy

Angles that share the same terminal side, created by adding or subtracting full rotations of 360° or 2π radians. 1. What Are Coterminal Angles? In standard position, angles start from the positive x-axis. As you rotate, the terminal side sweeps through the plane. If two angles end at the same position, they're coterminal—regardless of how many times you've rotated around the circle. 30° and 390° are coterminal because 390° = 30° + 360° Both angles end at the same position on the unit circle. To find a coterminal angle, add 360° (or 2π radians) to any angle. This represents one complete counterclockwise rotation that brings you back to the same terminal position. You can also subtract 360° (or 2π radians) to find coterminal angles. Subtracting represents rotating clockwise instead of counterclockwise. Negative angles still land at the same terminal position. Example: Finding a Negative Coterminal Angle Both 60° and -300° share the same terminal side. The negative angle represents rotating 300° clockwise from the positive x-axis. For any angle θ, you can generate infinitely many coterminal angles by adding or subtracting any integer multiple of 360° (or 2π radians). where n is any integer (..., -2, -1, 0, 1, 2, ...) Example: Multiple Coterminal Angles For θ = 30°, coterminal angles include: n = -1: 30° + 360°(-1) = -330° n = -2: 30° + 360°(-2) = -690° All of these angles share the same terminal side! 5. Finding Coterminal Angles in a Range

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