# What is the distance between the following polar coordinates?: # (2,(pi)/4), (5,(5pi)/8) #

If we denote

# theta = (5pi)/8- pi/4 = (3pi)/8 #

If we apply the cosine rule:

# a^2 = b^2 + c^2 - 2abcosA #

Then we get:

# (AB)^2 = (OA)^2 + (OB)^2 - 2(OA)(OB)costheta #

# " " = (2)^2 + (5)^2 - 2(2)(5)cos((3pi)/8) #

# " " = 4+25 - 20cos((3pi)/8) #

# " " = 29 - 20cos((3pi)/8) #

# " " = 21.34633... #

# :. AB = 4.629209 #

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

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

Given the polar coordinates ((r_1, \theta_1) = (2, \frac{\pi}{4})) and ((r_2, \theta_2) = (5, \frac{5\pi}{8})), we can plug in the values:

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

[ = \sqrt{4 + 25 - 20\cos\left(\frac{5\pi}{8} - \frac{\pi}{4}\right)} ]

[ = \sqrt{29 - 20\cos\left(\frac{5\pi}{8} - \frac{\pi}{4}\right)} ]

[ = \sqrt{29 - 20\cos\left(\frac{5\pi}{8} - \frac{2\pi}{8}\right)} ]

[ = \sqrt{29 - 20\cos\left(\frac{3\pi}{8}\right)} ]

Since ( \cos(\frac{3\pi}{8}) \approx 0.3827 ), we substitute this value:

[ d = \sqrt{29 - 20(0.3827)} ]

[ = \sqrt{29 - 7.654} ]

[ \approx \sqrt{21.346} ]

[ \approx 4.62 ]

So, the distance between the given polar coordinates is approximately ( 4.62 ).

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