# What is the distance between the following polar coordinates?: # (1,(4pi)/3), (4,(pi)/6) #

Plotting the coordinates, you may notice that the distance between the points and the origin form two sides of a triangle, and the distance between the two points themselves forms the third side.

Since we know the lengths of these two sides and the angle between them, we can use the Law of Cosines.

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

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

Given the coordinates:

[ (r_1, \theta_1) = \left(1, \frac{4\pi}{3}\right) ] [ (r_2, \theta_2) = \left(4, \frac{\pi}{6}\right) ]

Substituting into the formula:

[ \text{Distance} = \sqrt{(4^2 + 1^2 - 2 \cdot 4 \cdot 1 \cdot \cos(\frac{\pi}{6} - \frac{4\pi}{3}))} ]

[ \text{Distance} = \sqrt{(16 + 1 - 8 \cdot \cos(\frac{\pi}{6} - \frac{4\pi}{3}))} ]

[ \text{Distance} = \sqrt{(17 - 8 \cdot \cos(\frac{\pi}{6} - \frac{4\pi}{3}))} ]

[ \text{Distance} = \sqrt{(17 - 8 \cdot \cos(\frac{\pi}{6} - \frac{4\pi}{3}))} ]

[ \text{Distance} = \sqrt{(17 - 8 \cdot \cos(\frac{\pi}{6} - \frac{4\pi}{3}))} ]

[ \text{Distance} = \sqrt{(17 - 8 \cdot \cos(-\frac{5\pi}{6}))} ]

[ \text{Distance} = \sqrt{(17 - 8 \cdot (-\frac{\sqrt{3}}{2}))} ]

[ \text{Distance} = \sqrt{(17 + 4\sqrt{3})} ]

[ \text{Distance} = \sqrt{17 + 4\sqrt{3}} ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- What is the arclength of the polar curve #f(theta) = 2thetasin(5theta)-thetacot2theta # over #theta in [pi/12,pi/3] #?
- What is the distance between the following polar coordinates?: # (2,(7pi)/4), (7,(7pi)/8) #

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