# What is the distance between the following polar coordinates?:  (8,(5pi)/3), (1,(7pi)/4)

Jan 13, 2018

49.07 units

#### Explanation:

Formula for the distance between two points:

$D i s \tan c e = \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2}}$

Values:

$\Delta x = 8 - 1 = 7$

$\Delta y = \frac{5 \pi}{3} - \frac{7 \pi}{4} = - \frac{\pi}{12}$

Solve with equation

$D i s \tan c e = \sqrt{{\left(\Delta x\right)}^{2} + {\left(\Delta y\right)}^{2}}$

$D i s \tan c e = \sqrt{{\left(7\right)}^{2} + {\left(- \frac{\pi}{12}\right)}^{2}}$

$D i s \tan c e = 49.07 u n i t s$

Jan 13, 2018

$d \approx 7$

#### Explanation:

$\text{using the "color(blue)"polar version of the distance formula}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{{d}^{2} = {r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{2} - {\theta}_{1}\right)} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{let "(r_1,theta_1)=(8,(5pi)/3)" and } \left({r}_{2} , {\theta}_{2}\right) = \left(1 , \frac{7 \pi}{4}\right)$

${d}^{2} = {8}^{2} + {1}^{2} - 2.8 .1 \cos \left(\frac{7 \pi}{4} - \frac{5 \pi}{3}\right)$

$\textcolor{w h i t e}{{d}^{2}} = 65 - 16 \cos \left(\frac{\pi}{12}\right)$

$\Rightarrow d = \sqrt{65 - 16 \cos \left(\frac{\pi}{12}\right)} = 7.0388 \ldots$

$\Rightarrow d \approx 7 \text{ to nearest whole number}$