# What is the distance between the following polar coordinates?: # (1,(-5pi)/12), (8,(5pi)/8) #

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To find the distance between the given polar coordinates, we'll use the formula for the distance between two points in polar coordinates:

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

Substituting the given values:

( d = \sqrt{(8^2 + 1^2 - 2(8)(1)\cos\left(\frac{5\pi}{8} - \frac{-5\pi}{12}\right))} )

( d = \sqrt{(64 + 1 - 16\cos(\frac{5\pi}{8} + \frac{5\pi}{12}))} )

( d = \sqrt{(65 - 16\cos(\frac{15\pi}{24} + \frac{10\pi}{24}))} )

( d = \sqrt{(65 - 16\cos(\frac{25\pi}{24}))} )

( d ≈ \sqrt{(65 - 16\cos(\frac{\pi}{24}))} )

( d ≈ \sqrt{(65 - 16\cos(15))} )

( d ≈ \sqrt{(65 - 16 \cdot 0.9659)} )

( d ≈ \sqrt{(65 - 15.4544)} )

( d ≈ \sqrt{49.5456} )

( d ≈ 7.04 )

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