# How do you find the coterminal angle for (11pi) / 4?

Jul 15, 2015

Any angle of the form $\frac{11 \pi}{4} + 2 n \pi$ with $n \in \mathbb{Z}$ is coterminal with $\frac{11 \pi}{4}$

The coterminal angle of $\frac{11 \pi}{4}$ in $\left[0 , 2 \pi\right)$ is $\frac{3 \pi}{4}$

#### Explanation:

Coterminal angles are angles which are equal modulo $2 \pi$

That is: $\alpha$ and $\beta$ are coterminal angles if $\alpha - \beta = 2 n \pi$ for some integer $n$.

For example, $\frac{11 \pi}{4}$ and $\frac{3 \pi}{4}$ are coterminal, since:

$\frac{11 \pi}{4} - \frac{3 \pi}{4} = \frac{8 \pi}{4} = 2 \pi = 2 n \pi$ with $n = 1$

Every angle has a unique coterminal angle in the range $\left[0 , 2 \pi\right)$

If $\theta \ge 0$ then $\theta - 2 \left\lfloor \frac{\theta}{2 \pi} \right\rfloor \pi \in \left[0 , 2 \pi\right)$

If $\theta < 0$ then $\theta + 2 \left\lceil \frac{- \theta}{2 \pi} \right\rceil \pi \in \left[0 , 2 \pi\right)$

Coterminality is an example of an equivalence relation

If we use the symbol ~ to mean "is coterminal with" then we find:

Reflexive: For all $\alpha$: alpha ~ alpha

Commutative: For all $\alpha , \beta$: alpha ~ beta <=> beta ~ alpha

Transitive: For all $\alpha$, $\beta$, $\gamma$: if alpha ~ beta and beta ~ gamma then alpha ~ gamma