# What is the distance between the following polar coordinates?: # (12,(13pi)/8), (19,(-7pi)/8) #

The distance formula for polar coordinates

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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 ( (12, \frac{13\pi}{8}) ) and ( (19, \frac{-7\pi}{8}) ), let's plug in the values:

[ r_1 = 12, \ r_2 = 19, \ \theta_1 = \frac{13\pi}{8}, \ \theta_2 = \frac{-7\pi}{8} ]

[ d = \sqrt{(12)^2 + (19)^2 - 2(12)(19)\cos\left(\frac{-7\pi}{8} - \frac{13\pi}{8}\right)} ]

Now, calculate the cosine term:

[ \cos\left(\frac{-7\pi}{8} - \frac{13\pi}{8}\right) = \cos\left(\frac{-20\pi}{8}\right) = \cos\left(\frac{-5\pi}{2}\right) = 0 ]

So, the distance simplifies to:

[ d = \sqrt{(12)^2 + (19)^2 - 2(12)(19)(0)} ]

[ d = \sqrt{144 + 361 - 0} ]

[ d = \sqrt{505} ]

Therefore, the distance between the given polar coordinates is ( \sqrt{505} ).

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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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