# What is the distance between the following polar coordinates?: # (4,(-8pi)/3), (-5,(11pi)/6) #

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Accordingly, the reqd. dist.

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To find the distance between the polar coordinates (4, (-8π)/3) and (-5, (11π)/6), we use the formula for the distance between two points in polar coordinates:

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

where ( r_1 ) and ( r_2 ) are the magnitudes (or lengths) of the two vectors and ( \theta_1 ) and ( \theta_2 ) are their respective angles.

Given the coordinates (r, θ), the magnitudes and angles are as follows:

For (4, (-8π)/3): ( r_1 = 4 ) and ( \theta_1 = -8π/3 ).

For (-5, (11π)/6): ( r_2 = -5 ) and ( \theta_2 = 11π/6 ).

Now, we substitute these values into the formula:

[ \text{Distance} = \sqrt{4^2 + (-5)^2 - 2(4)(-5)\cos\left(\frac{11\pi}{6} + \frac{8\pi}{3}\right)} ]

Simplify the angles inside the cosine function:

[ \frac{11\pi}{6} + \frac{8\pi}{3} = \frac{22\pi}{6} + \frac{16\pi}{6} = \frac{38\pi}{6} = \frac{19\pi}{3} ]

[ \text{Distance} = \sqrt{16 + 25 + 40\cos\left(\frac{19\pi}{3}\right)} ]

Since cosine is periodic with period ( 2\pi ), ( \cos\left(\frac{19\pi}{3}\right) = \cos\left(\frac{19\pi}{3} - 6\pi\right) = \cos\left(\frac{1\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} ).

[ \text{Distance} = \sqrt{41 + 40\cdot\frac{1}{2}} ]

[ \text{Distance} = \sqrt{41 + 20} ]

[ \text{Distance} = \sqrt{61} ]

Thus, the distance between the polar coordinates (4, (-8π)/3) and (-5, (11π)/6) is ( \sqrt{61} ).

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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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