# What is the distance between the following polar coordinates?: # (3,(-7pi)/12), (7,(5pi)/8) #

The distance formula for polar coordinates is

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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)} ]

Plugging in the values:

[ r_1 = 3, \theta_1 = -\frac{7\pi}{12}, r_2 = 7, \theta_2 = \frac{5\pi}{8} ]

we get:

[ d = \sqrt{3^2 + 7^2 - 2 \cdot 3 \cdot 7 \cdot \cos\left(\frac{5\pi}{8} + \frac{7\pi}{12}\right)} ]

Simplify the angles inside the cosine:

[ \frac{5\pi}{8} + \frac{7\pi}{12} = \frac{15\pi}{24} + \frac{14\pi}{24} = \frac{29\pi}{24} ]

Then, calculate the cosine:

[ \cos\left(\frac{29\pi}{24}\right) = \cos\left(\frac{24\pi}{24} + \frac{5\pi}{24}\right) = \cos\left(\frac{5\pi}{24}\right) ]

Now, calculate the distance:

[ d = \sqrt{9 + 49 - 42\cos\left(\frac{5\pi}{24}\right)} ]

[ d = \sqrt{58 - 42\cos\left(\frac{5\pi}{24}\right)} ]

This is the exact expression for the distance between the two polar coordinates.

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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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