# What is the distance between the following polar coordinates?: # (7,(5pi)/4), (2,(9pi)/8) #

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To find the distance between two polar coordinates, we can use the distance formula derived from the Pythagorean theorem. The formula is:

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

Given the coordinates (7, (\frac{5\pi}{4})) and (2, (\frac{9\pi}{8})), let's denote: ( r_1 = 7 ), ( r_2 = 2 ), ( \theta_1 = \frac{5\pi}{4} ), ( \theta_2 = \frac{9\pi}{8} ).

Now, we can plug these values into the distance formula:

[ \text{Distance} = \sqrt{(2^2 + 7^2 - 2 \cdot 2 \cdot 7 \cdot \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

[ \text{Distance} = \sqrt{(4 + 49 - 28 \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(\frac{9\pi}{8} - \frac{5\pi}{4}))} ]

Now, let's calculate the difference of the angles: [ \frac{9\pi}{8} - \frac{5\pi}{4} = \frac{9\pi}{8} - \frac{10\pi}{8} = -\frac{\pi}{8} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(-\frac{\pi}{8}))} ]

[ \text{Distance} = \sqrt{(53 - 28 \cos(\frac{\pi}{8}))} ]

Now, we can calculate the cosine of (\frac{\pi}{8}) which is approximately ( \cos(\frac{\pi}{8}) \approx 0.92388 ).

[ \text{Distance} = \sqrt{(53 - 28 \times 0.92388)} ]

[ \text{Distance} = \sqrt{(53 - 25.85344)} ]

[ \text{Distance} = \sqrt{(27.14656)} ]

[ \text{Distance} \approx 5.21 ]

So, the distance between the polar coordinates is approximately 5.21 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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