fabios3

2022-06-24

What is the distance between the following polar coordinates?:

$(5,\frac{7\pi}{4}),(9,\frac{11\pi}{8})$

$(5,\frac{7\pi}{4}),(9,\frac{11\pi}{8})$

last99erib

Beginner2022-06-25Added 19 answers

Step 1

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

$d=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}-2{r}_{1}{r}_{2}\mathrm{cos}({\theta}_{2}-{\theta}_{1})}$

$\text{Given}\phantom{\rule{1ex}{0ex}}{r}_{1}=5,{r}_{2}=9,{\theta}_{1}={\left(\frac{7\pi}{4}\right)}^{c},{\theta}_{2}={\left(\frac{11\pi}{8}\right)}^{c}$

$d=\sqrt{{5}^{2}+{9}^{2}-(2\cdot 5\cdot 9\cdot \mathrm{cos}(\frac{11\pi}{8}-\frac{7\pi}{4}))}$

$\textcolor[rgb]{}{d}=\sqrt{106-90\mathrm{cos}\left(\frac{-3\pi}{8}\right)}\textcolor[rgb]{}{\approx 8.4592}$

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

$d=\sqrt{{r}_{1}^{2}+{r}_{2}^{2}-2{r}_{1}{r}_{2}\mathrm{cos}({\theta}_{2}-{\theta}_{1})}$

$\text{Given}\phantom{\rule{1ex}{0ex}}{r}_{1}=5,{r}_{2}=9,{\theta}_{1}={\left(\frac{7\pi}{4}\right)}^{c},{\theta}_{2}={\left(\frac{11\pi}{8}\right)}^{c}$

$d=\sqrt{{5}^{2}+{9}^{2}-(2\cdot 5\cdot 9\cdot \mathrm{cos}(\frac{11\pi}{8}-\frac{7\pi}{4}))}$

$\textcolor[rgb]{}{d}=\sqrt{106-90\mathrm{cos}\left(\frac{-3\pi}{8}\right)}\textcolor[rgb]{}{\approx 8.4592}$