# What is the distance between the following polar coordinates?:  (1,(4pi)/3), (4,(pi)/6)

Nov 4, 2017

$\sqrt{17 + 4 \sqrt{3}} \approx 4.89$

#### Explanation:

Plotting the coordinates, you may notice that the distance between the points and the origin form two sides of a triangle, and the distance between the two points themselves forms the third side.

Since we know the lengths of these two sides and the angle between them, we can use the Law of Cosines.

The angle is $\frac{4 \pi}{3} - \frac{\pi}{6} = \frac{8 \pi}{6} - \frac{\pi}{6} = \frac{7 \pi}{6}$, and the side lengths are $1$ and $4$.

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos C$

${c}^{2} = {1}^{2} + {4}^{2} - \left[2 \cdot 1 \cdot 4 \cdot \cos \left(\frac{7 \pi}{6}\right)\right]$

${c}^{2} = 17 - \left[8 \cos \left(\frac{7 \pi}{6}\right)\right]$

${c}^{2} = 17 - \left(8 \cdot - \frac{\sqrt{3}}{2}\right)$

${c}^{2} = 17 + 4 \sqrt{3}$

$c = \sqrt{17 + 4 \sqrt{3}} \approx 4.89$