# What is the distance between the following polar coordinates?: # (2,(pi)/8), (7,(3pi)/8) #

As scene of the picture, i have drawn each of the polar coordinates, with there respective lengths and angles

We can see that

Now we can use the cosine rule to find

As

Hence yielding:

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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)} ]

Substitute the given values:

( r_1 = 2 ), ( \theta_1 = \frac{\pi}{8} )

( r_2 = 7 ), ( \theta_2 = \frac{3\pi}{8} )

Then, plug these values into the formula and calculate:

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

[ d = \sqrt{4 + 49 - 28\cos\left(\frac{\pi}{2}\right)} ]

[ d = \sqrt{53 - 0} ]

[ d = \sqrt{53} ]

So, the distance between the polar coordinates ( (2, \frac{\pi}{8}) ) and ( (7, \frac{3\pi}{8}) ) is ( \sqrt{53} ) units.

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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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