# What is the distance between the following polar coordinates?:  (2,(pi)/2), (2,(17pi)/12)

Jan 1, 2018

$\overline{A B} \setminus \approx 3.9658$

#### Explanation:

Graph the two points:

A triangle is formed when the two points $A$ and $B$ are connected to the origin $O$. We know the angle $\angle A O B$ is $\frac{17 \pi}{12} - \frac{\pi}{2} = \frac{11 \pi}{12}$, and the two sides $\overline{A O} = \overline{B O} = 2$.

Then, $\overline{A B}$ can be found using the law of cosines:
${\overline{A B}}^{2} = {\overline{A O}}^{2} + {\overline{B O}}^{2} - 2 \cdot \overline{A O} \cdot \overline{B O} \cdot \cos \left(\angle A O B\right)$
${\overline{A B}}^{2} = {2}^{2} + {2}^{2} - 2 \cdot 2 \cdot 2 \cdot \cos \left(\frac{11 \pi}{12}\right)$
${\overline{A B}}^{2} \setminus \approx 15.7274$

$\therefore \overline{A B} \setminus \approx 3.9658$