Rapsinincke

2022-07-04

What is the distance between the following polar coordinates?: $\left(3,\frac{\pi }{4}\right),\left(9,\frac{3\pi }{8}\right)$

Expert

Step 1
If the points are and and O is the pole $r=0$ , then
Length $PQ=\sqrt{O{Q}^{2}+G{P}^{2}-2\left(OP\right)\left(OQ\right)\mathrm{cos}\angle QOP}$
$=\sqrt{{9}^{2}+{2}^{2}-2\left(9\right)\left(2\right)\mathrm{cos}\left(\frac{3}{8}\pi -\frac{\pi }{4}\right)}$
$=\sqrt{85-36\mathrm{cos}\left(\frac{\pi }{8}\right)}$
$=\sqrt{85-36\sqrt{\left(\frac{1-\mathrm{cos}\left(\frac{\pi }{4}\right)}{2}\right)}}$
$=\sqrt{85-36\sqrt{\frac{1-\frac{1}{\sqrt{2}}}{2}}}\right)$
$=8.4394$, nearly.

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