What is the distance between the following polar coordinates?:  (7,(5pi)/4), (2,(9pi)/8)

Apr 4, 2016

${P}_{1} {P}_{2} = \sqrt{53 - 28 \cos \left(\frac{\pi}{8}\right)} \approx 5.209$

Explanation:

${P}_{1} {P}_{2} = \sqrt{{r}_{1}^{2} + {r}_{2}^{2} - 2 {r}_{1} {r}_{2} \cos \left({\theta}_{2} - {\theta}_{1}\right)}$
r_1 = 7, theta_1 =(5pi)/4;r_2 =2, theta_2 =(9pi)/8
${P}_{1} {P}_{2} = \sqrt{{7}^{2} + {2}^{2} - 2 \cdot 7 \cdot 2 \cos \left(\frac{9 \pi}{8} - \frac{5 \pi}{4}\right)}$
P_1P_2 = sqrt(49+4-28cos(-(pi)/8)
${P}_{1} {P}_{2} = \sqrt{53 - 28 \cos \left(\frac{\pi}{8}\right)} \approx 5.209$

Apr 4, 2016

$s \cong 5 , 27$

Explanation:

${r}_{1} = 7$
${r}_{2} = 2$
${\theta}_{1} = \frac{5 \pi}{4}$
${\theta}_{2} = \frac{9 \pi}{8}$
${\theta}_{2} - {\theta}_{1} = \frac{9 \pi}{8} - \frac{5 \pi}{4} = \frac{9 \pi - 10 \pi}{8} = - \frac{\pi}{8}$

$\cos \left(- \frac{\pi}{8}\right) = 0 , 9$

$s = \sqrt{{r}_{1}^{2} + {r}_{2}^{2} - 2 \cdot {r}_{1} \cdot {r}_{2} \cdot \cos \left({\theta}_{2} - {\theta}_{1}\right)}$

$s = \sqrt{{7}^{2} + {2}^{2} - 2 \cdot 7 \cdot 2 \cdot 0 , 9}$

$s = \sqrt{49 + 4 - 28 \cdot 0 , 9}$

$s = \sqrt{53 - 25.2}$

$s = \sqrt{27 , 8}$
$s \cong 5.27$