# What is the distance between the following polar coordinates?: # (3,(5pi)/4), (1,(pi)/8) #

Distance between tow points knowing the polar coordinates is given by the formula using cosine rule

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To find the distance between the polar coordinates ((3, \frac{5\pi}{4})) and ((1, \frac{\pi}{8})), you can use the formula for the distance (d) between two points ((r_1, \theta_1)) and ((r_2, \theta_2)) in polar coordinates:

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

Substitute the given values:

( r_1 = 3 ), ( \theta_1 = \frac{5\pi}{4} )

( r_2 = 1 ), ( \theta_2 = \frac{\pi}{8} )

[ d = \sqrt{3^2 + 1^2 - 2(3)(1) \cos\left(\frac{\pi}{8} - \frac{5\pi}{4}\right)} ]

Now, calculate the angle difference and the cosine value:

[ \frac{\pi}{8} - \frac{5\pi}{4} = -\frac{15\pi}{8} ]

[ \cos\left(-\frac{15\pi}{8}\right) = \cos\left(\frac{\pi}{8}\right) ]

Finally, substitute back into the distance formula:

[ d = \sqrt{9 + 1 - 6 \cos\left(\frac{\pi}{8}\right)} ]

Evaluate ( \cos\left(\frac{\pi}{8}\right) ) and then calculate the distance ( d ).

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