# What is the distance between the following polar coordinates?: # (4,(-7pi)/12), (2,(pi)/8) #

The distance formula for polar coordinates can be derived from the distance formula for rectangular coordinates

Plugging those and using a couple of trigonometric identities, you get the following in purely polar coordinates

Plugging in the polar coordinates you have been given, we get

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To find the distance between two polar coordinates ( (r_1, \theta_1) ) and ( (r_2, \theta_2) ), you can use the formula:

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

Using the given polar coordinates:

( r_1 = 4 ), ( \theta_1 = -\frac{7\pi}{12} )

( r_2 = 2 ), ( \theta_2 = \frac{\pi}{8} )

Substituting these values into the formula:

[ \begin{split} d & = \sqrt{(4)^2 + (2)^2 - 2(4)(2)\cos\left(\frac{\pi}{8} - \left(-\frac{7\pi}{12}\right)\right)} \ & = \sqrt{16 + 4 - 16\cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right)} \ & = \sqrt{20 - 16\cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right)} \end{split} ]

To evaluate ( \cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right) ):

[ \begin{split} \cos\left(\frac{\pi}{8} + \frac{7\pi}{12}\right) & = \cos\left(\frac{3\pi}{24} + \frac{14\pi}{24}\right) \ & = \cos\left(\frac{17\pi}{24}\right) \end{split} ]

Since ( \cos ) is periodic with a period of ( 2\pi ), we can subtract ( 2\pi ) to bring the angle within one full period:

[ \begin{split} \cos\left(\frac{17\pi}{24}\right) & = \cos\left(\frac{17\pi}{24} - 2\pi\right) \ & = \cos\left(\frac{17\pi}{24} - \frac{48\pi}{24}\right) \ & = \cos\left(-\frac{31\pi}{24}\right) \end{split} ]

Since ( \cos(-x) = \cos(x) ), we have:

[ \cos\left(-\frac{31\pi}{24}\right) = \cos\left(\frac{31\pi}{24}\right) ]

So, ( d = \sqrt{20 - 16\cos\left(\frac{31\pi}{24}\right)} ). You can approximate this value numerically.

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