# What is the distance between the following polar coordinates?: # (2,(pi)/3), (4,(pi)/6) #

distance

Find the distance between the two points:

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

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

Given the polar coordinates ( (2, \frac{\pi}{3}) ) and ( (4, \frac{\pi}{6}) ), we have:

[ r_1 = 2, \quad \theta_1 = \frac{\pi}{3}, \quad r_2 = 4, \quad \theta_2 = \frac{\pi}{6} ]

Substituting these values into the formula:

[ d = \sqrt{2^2 + 4^2 - 2(2)(4)\cos\left(\frac{\pi}{6} - \frac{\pi}{3}\right)} ]

[ d = \sqrt{4 + 16 - 16\cos\left(\frac{\pi}{6} - \frac{\pi}{3}\right)} ]

[ d = \sqrt{20 - 16\cos\left(\frac{\pi}{6} - \frac{\pi}{3}\right)} ]

[ d = \sqrt{20 - 16\cos\left(-\frac{\pi}{6}\right)} ]

[ d = \sqrt{20 - 16\left(\frac{\sqrt{3}}{2}\right)} ]

[ d = \sqrt{20 - 8\sqrt{3}} ]

Thus, the distance between the polar coordinates ( (2, \frac{\pi}{3}) ) and ( (4, \frac{\pi}{6}) ) is ( \sqrt{20 - 8\sqrt{3}} ).

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