Coterminal Angles
TrigonometryCoterminal angles are angles in standard position that share the same terminal side, differing by full rotations of 360 degrees 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 that end up in the same position after rotation. They differ by one or more full turns ($360^\circ$).
Example
$45^\circ$ and $405^\circ$ are coterminal because $405^\circ = 45^\circ + 360^\circ$. They both point to the same place. Also, $-315^\circ$ is coterminal with $45^\circ$.
Key Insight
Spinning around one full circle brings you back to where you started. Any angle plus or minus $360^\circ$ lands on the same spot.
Definition
Two angles are coterminal if they have the same terminal side in standard position. Adding or subtracting any multiple of $360^\circ$ (or $2\pi$ radians) produces a coterminal angle. To find a positive coterminal angle less than $360^\circ$, add or subtract $360^\circ$ until in range.
Example
Find coterminals of $750^\circ$: $750^\circ - 360^\circ = 390^\circ$; $390^\circ - 360^\circ = 30^\circ$. So $750^\circ$, $390^\circ$, and $30^\circ$ are all coterminal. Also, $750^\circ - 720^\circ = 30^\circ$ directly.
Key Insight
Because coterminal angles share a terminal side, they have identical trig values: $\sin(750^\circ) = \sin(30^\circ) = 0.5$. This is just a restatement of the periodicity: $\sin(\theta + 2\pi n) = \sin(\theta)$.
Definition
Coterminal angles are elements of the same equivalence class in $\mathbb{R}/(2\pi\mathbb{Z})$: two angles are coterminal iff their difference is a multiple of $2\pi$. The set of all angles coterminal with $\theta$ is $\{\theta + 2\pi k : k \in \mathbb{Z}\}$. All trig functions are $2\pi$-periodic, so they are well-defined on the quotient space $\mathbb{R}/(2\pi\mathbb{Z}) = S^1$.
Example
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 $2\pi\mathbb{Z}$. The fundamental domain is $[0, 2\pi)$ or equivalently $(-\pi, \pi]$.
Key Insight
The quotient $\mathbb{R}/(2\pi\mathbb{Z})$ is topologically a circle. This is the abstract version of the statement that angles wrap around. The same structure appears in the theory of periodic functions, Fourier series, and in the definition of the fundamental group $\pi_1(S^1) = \mathbb{Z}$.