fabios3

2022-06-24

What is the distance between the following polar coordinates?:
$\left(5,\frac{7\pi }{4}\right),\left(9,\frac{11\pi }{8}\right)$

last99erib

Step 1
Distance between tow points knowing the polar coordinates is given by the formula using cosine rule
$d=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}-2{r}_{1}{r}_{2}\mathrm{cos}\left({\theta }_{2}-{\theta }_{1}\right)}$
$\text{Given}\phantom{\rule{1ex}{0ex}}{r}_{1}=5,{r}_{2}=9,{\theta }_{1}={\left(\frac{7\pi }{4}\right)}^{c},{\theta }_{2}={\left(\frac{11\pi }{8}\right)}^{c}$
$d=\sqrt{{5}^{2}+{9}^{2}-\left(2\cdot 5\cdot 9\cdot \mathrm{cos}\left(\frac{11\pi }{8}-\frac{7\pi }{4}\right)\right)}$
$d=\sqrt{106-90\mathrm{cos}\left(\frac{-3\pi }{8}\right)}\approx 8.4592$

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