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If a snowboarder completes a 360Β° spin and another pulls off a 720Β° spin, they both land facing the exact same direction down the mountain.
Even though one spun around twice as many times, their starting and ending orientations are completely identical.

What Are Coterminal Angles?

In trigonometry, an angle in standard position places its vertex at the origin (0,0)(0,0) and its fixed initial side along the positive xx-axis.
The rotating ray is the terminal side, and coterminal angles are angles in standard position that share the exact same terminal side.
Because a complete circle measures exactly 360∘360^\circ, rotating one full turn in either direction lands you right back where you started.
To find any coterminal angle for an angle ΞΈ\theta, use the coterminal formula:
ΞΈcoterminal=ΞΈ+kβ‹…360∘\theta_{\text{coterminal}} = \theta + k \cdot 360^\circ where kk is any integer (…,βˆ’2,βˆ’1,0,1,2,…\dots, -2, -1, 0, 1, 2, \dots).
What if you need to find specific coterminal angles for a given angle on an exam? Let's work through a direct example.

Finding Positive and Negative Coterminal Angles

Suppose you are given θ=110∘\theta = 110^\circ and asked to find one positive and one negative coterminal angle.
To find a positive coterminal angle, add 360∘360^\circ (k=1k = 1):