# What is the distance between #(4 ,( 9 pi)/8 )# and #(-1 ,( 3 pi )/2 )#?

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To find the distance between two points in polar coordinates, we can use the formula:

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

Given the points ((4, \frac{9\pi}{8})) and ((-1, \frac{3\pi}{2})), we have:

[ r_1 = 4, \quad r_2 = -1, \quad \theta_1 = \frac{9\pi}{8}, \quad \theta_2 = \frac{3\pi}{2} ]

[ \text{Distance} = \sqrt{(-1)^2 + (4)^2 - 2(-1)(4) \cos(\frac{3\pi}{2} - \frac{9\pi}{8})} ]

[ = \sqrt{1 + 16 + 8\cos(\frac{3\pi}{2} - \frac{9\pi}{8})} ]

[ = \sqrt{17 + 8\cos(\frac{12\pi - 9\pi}{8})} ]

[ = \sqrt{17 + 8\cos(\frac{3\pi}{8})} ]

[ = \sqrt{17 + 8 \left(\frac{\sqrt{2 + \sqrt{2}}}{2}\right)} ]

[ = \sqrt{17 + 4\sqrt{2 + \sqrt{2}}} ]

[ \approx \sqrt{22.657} ]

[ \approx 4.757 ]

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