# What is the distance between the following polar coordinates?:  (12,(pi)/8), (-8,(5pi)/8)

Jul 9, 2018

color(crimson)("Distance " d ~~ 14.4222

#### Explanation:

Distance between tow points knowing the polar coordinates is given by the formula using cosine rule

d = sqrt(r_1 ^2 + r_2 ^2 - 2 r_1 r_2 cos (theta_2 - theta_1)

$\text{Given } {r}_{1} = 12 , {r}_{2} = - 8 , {\theta}_{1} = {\left(\frac{\pi}{8}\right)}^{c} , {\theta}_{2} = {\left(\frac{5 \pi}{8}\right)}^{c}$

d = sqrt (12^2 + -8^2 - (2 * 12 * -8 * cos ((5pi)/8 - (pi)/8))

$\textcolor{c r i m s o n}{d} = \sqrt{208 + \cancel{192 \cos \left(\frac{\pi}{2}\right)}} \textcolor{c r i m s o n}{\approx 14.4222} , \text{ as } \cos \left(\frac{\pi}{2}\right) = 0$