# What is the distance between the following polar coordinates?: # (4,(-11pi)/12), (1,(-7pi)/8) #

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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)}]

Given the polar coordinates ((4, -\frac{11\pi}{12})) and ((1, -\frac{7\pi}{8})), we have:

[r_1 = 4, \quad \theta_1 = -\frac{11\pi}{12}] [r_2 = 1, \quad \theta_2 = -\frac{7\pi}{8}]

Plugging these values into the formula:

[d = \sqrt{(4)^2 + (1)^2 - 2(4)(1) \cos\left(-\frac{7\pi}{8} - \left(-\frac{11\pi}{12}\right)\right)}]

[d = \sqrt{16 + 1 - 8\cos\left(-\frac{7\pi}{8} + \frac{11\pi}{12}\right)}]

[d = \sqrt{17 - 8\cos\left(\frac{11\pi}{12} - \frac{7\pi}{8}\right)}]

[d = \sqrt{17 - 8\cos\left(\frac{11\pi}{12} - \frac{21\pi}{24}\right)}]

[d = \sqrt{17 - 8\cos\left(\frac{\pi}{24}\right)}]

Since (\cos\left(\frac{\pi}{24}\right)) is a known value, we can substitute it and compute the distance:

[d = \sqrt{17 - 8\cos\left(\frac{\pi}{24}\right)}] [d = \sqrt{17 - 8\sqrt{\frac{1 + \sqrt{3}}{8}}}]

[d \approx \sqrt{17 - 4\sqrt{1 + \sqrt{3}}}]

Therefore, the distance between the given polar coordinates is approximately (\sqrt{17 - 4\sqrt{1 + \sqrt{3}}}).

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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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