Coterminal Angles
Coterminal angles are angles in standard position that share the same terminal side, differing by full rotations of $360^\circ$ or $2\pi$ radians.
Formula
\text{coterminal angles} = \theta + 360n \text{ degrees (or } \theta + 2\pi n \text{ radians) for any integer } n
Definition
Coterminal angles are angles in standard position that share the same terminal side, differing by one or more full rotations of $360^\circ$ (or $2\pi$ radians). Two angles are coterminal if and only if their difference is a multiple of $2\pi$, so the set of angles coterminal with $\theta$ is $\{\theta + 2\pi k : k \in \mathbb{Z}\}$; formally, they are elements of the same equivalence class in $\mathbb{R}/(2\pi\mathbb{Z})$, and because all trig functions are $2\pi$-periodic, they are well-defined on this quotient space, which is topologically a circle.
Example
$45^\circ$ and $405^\circ$ are coterminal because $405^\circ = 45^\circ + 360^\circ$, and $-315^\circ$ is also coterminal with $45^\circ$. To find a positive coterminal angle of $750^\circ$ less than $360^\circ$: $750^\circ - 360^\circ = 390^\circ$, then $390^\circ - 360^\circ = 30^\circ$. In complex analysis, $e^{i\theta} = e^{i(\theta + 2\pi k)}$ for all integers $k$, confirming that the exponential map $e^{i\theta}: \mathbb{R} \to S^1$ has kernel $2\pi\mathbb{Z}$, with fundamental domain $[0, 2\pi)$ or equivalently $(-\pi, \pi]$.
Key Insight
Spinning around one full circle brings you back to where you started, so any angle plus or minus $360^\circ$ lands on the same spot, and coterminal angles share identical trig values, $\sin(750^\circ) = \sin(30^\circ) = 0.5$, just a restatement of periodicity. The quotient $\mathbb{R}/(2\pi\mathbb{Z})$ is the abstract version of the statement that angles wrap around; the same structure appears in Fourier series and in the definition of the fundamental group $\pi_1(S^1) = \mathbb{Z}$.