# What is the distance between the following polar coordinates?:  (8,(7pi)/4), (5,(15pi)/8)

Nov 1, 2016

The distance is $\approx 3.88$

#### Explanation:

The origin and these two points:

$\left(8 , \frac{7 \pi}{7}\right) \mathmr{and} \left(5 , \frac{15 \pi}{8}\right)$

Form a triangle with,

side $a = 8$,
side $b = 5$,

and the angle between them is,

$C = \frac{15 \pi}{8} - \frac{7 \pi}{4} = \frac{\pi}{8}$.

The distance between the two points is the length of side c, in the equation for the Law of Cosines:

${c}^{2} = {a}^{2} + {b}^{2} - 2 \left(a\right) \left(b\right) \cos \left(C\right)$

${c}^{2} = {8}^{2} + {5}^{2} - 2 \left(8\right) \left(5\right) \cos \left(\frac{\pi}{8}\right)$

$c \approx 3.88$