Enter any angle in degrees or radians to find its principal angle, positive and negative co-terminal angles, the general formula, and an interactive unit circle diagram showing multiple co-terminal rays.
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Concept 1 — What Are Co-terminal Angles?
θ and θ ± 360°k (k ∈ ℤ)
Two angles are co-terminal when they share the same terminal ray — the ray that the angle's rotation lands on in standard position (vertex at origin, initial side along the positive x-axis).
You can add or subtract any whole-number multiple of 360° (one full rotation) and arrive at the exact same direction. In radians, the period is 2π.
For example, 30°, 390°, 750°, and −330° all point in the same direction — they are all co-terminal with each other.
Concept 2 — Why Trig Functions Are Periodic
sin(θ) = sin(θ + 360°·n)
Because sine, cosine, and the other trig functions are defined by the coordinates on the unit circle, and co-terminal angles hit the same point on the circle, every trig function returns the identical value at co-terminal angles.
This is exactly what "period 360°" means: the function's output repeats after one full revolution.
A Naruhodo tutor can walk you through standard position, co-terminal angles, and the entire unit circle in a live one-on-one session.