# What is the distance between the following polar coordinates?: # (5,(5pi)/4), (2,(11pi)/8) #

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The distance between the polar coordinates (5, ( \frac{5\pi}{4} )) and (2, ( \frac{11\pi}{8} )) can be calculated using the formula for the distance between two points in polar coordinates:

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

Substitute the given values:

[ d = \sqrt{5^2 + 2^2 - 2 \cdot 5 \cdot 2 \cdot \cos\left(\frac{11\pi}{8} - \frac{5\pi}{4}\right)} ]

Simplify and compute the cosine term:

[ d = \sqrt{25 + 4 - 20\cos\left(\frac{3\pi}{8}\right)} ]

Calculate the cosine value and simplify further:

[ d = \sqrt{29 - 20\cos\left(\frac{3\pi}{8}\right)} ]

Using trigonometric identities and calculations, we find the cosine value and then the final distance:

[ \cos\left(\frac{3\pi}{8}\right) \approx -0.3827 ]

[ d \approx \sqrt{29 - 20(-0.3827)} ]

[ d \approx \sqrt{29 + 7.654} ]

[ d \approx \sqrt{36.654} ]

[ d \approx 6.05 ]

So, the distance between the given polar coordinates is approximately 6.05 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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