What is the distance between the following polar coordinates?: # (2,(3pi)/4), (9,(13pi)/8) #

To find the distance between the polar coordinates ( (r_1, \theta_1) = (2, \frac{3\pi}{4}) ) and ( (r_2, \theta_2) = (9, \frac{13\pi}{8}) ), we use the formula:

[ d = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_2 - \theta_1)} ]

Substituting the given values:

[ d = \sqrt{2^2 + 9^2 - 2(2)(9)\cos\left(\frac{13\pi}{8} - \frac{3\pi}{4}\right)} ]

[ d = \sqrt{4 + 81 - 36\cos\left(\frac{13\pi}{8} - \frac{3\pi}{4}\right)} ]

[ d = \sqrt{85 - 36\cos\left(\frac{13\pi}{8} - \frac{3\pi}{4}\right)} ]

Now, calculate ( \frac{13\pi}{8} - \frac{3\pi}{4} = \frac{\pi}{8} ).

[ d = \sqrt{85 - 36\cos\left(\frac{\pi}{8}\right)} ]

[ d = \sqrt{85 - 36\cos\left(\frac{\pi}{8}\right)} ]

[ d ≈ \sqrt{85 - 36(0.9239)} ]

[ d ≈ \sqrt{85 - 33.3218} ]

[ d ≈ \sqrt{51.6782} ]

[ d ≈ 7.1871 ]

Therefore, the distance between the given polar coordinates is approximately ( 7.1871 ).