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

Distance between tow points knowing the polar coordinates is given by the formula using cosine rule

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

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

Given:

- ( r_1 = 12 )
- ( \theta_1 = \frac{\pi}{8} )
- ( r_2 = -8 )
- ( \theta_2 = \frac{5\pi}{8} )

Plugging in the values:

[ \text{Distance} = \sqrt{(12)^2 + (-8)^2 - 2(12)(-8)\cos\left(\frac{5\pi}{8} - \frac{\pi}{8}\right)} ]

[ = \sqrt{144 + 64 + 192\cos\left(\frac{5\pi}{8} - \frac{\pi}{8}\right)} ]

[ = \sqrt{208 + 192\cos\left(\frac{4\pi}{8}\right)} ]

[ = \sqrt{208 + 192\cos\left(\frac{\pi}{2}\right)} ]

[ = \sqrt{208} ]

[ = 4\sqrt{13} ]

So, the distance between the given polar coordinates is ( 4\sqrt{13} ).

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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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