# What is the distance between the following polar coordinates?: # (6,(7pi)/12), (2,(-3pi)/8) #

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

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

Plugging in the given values:

[ r_1 = 6, \theta_1 = \frac{7\pi}{12}, r_2 = 2, \theta_2 = -\frac{3\pi}{8} ]

[ \text{Distance} = \sqrt{(6^2 + 2^2 - 2(6)(2)\cos\left(\frac{7\pi}{12} - (-\frac{3\pi}{8})\right))} ]

[ = \sqrt{(36 + 4 - 24\cos(\frac{7\pi}{12} + \frac{3\pi}{8}))} ]

[ = \sqrt{(40 - 24\cos(\frac{7\pi}{12} + \frac{3\pi}{8}))} ]

Since (\cos(\frac{7\pi}{12} + \frac{3\pi}{8})) cannot be simplified further without using trigonometric identities, we'll leave it in this form. Thus, the distance between the given polar coordinates is:

[ \text{Distance} = \sqrt{(40 - 24\cos(\frac{7\pi}{12} + \frac{3\pi}{8}))} ]

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