# What is the distance between the following polar coordinates?: # (8,(7pi)/4), (5,(15pi)/8) #

The distance is

The origin and these two points:

Form a triangle with,

and the angle between them is,

The distance between the two points is the length of side c, in the equation for the Law of Cosines:

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To find the distance between two polar coordinates, we can use the formula:

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

Given the coordinates:

[r_1 = 8, \theta_1 = \frac{7\pi}{4}]

[r_2 = 5, \theta_2 = \frac{15\pi}{8}]

Plugging in these values, we get:

[d = \sqrt{8^2 + 5^2 - 2*8*5*\cos\left(\frac{15\pi}{8} - \frac{7\pi}{4}\right)}]

Simplifying the angle difference:

[\frac{15\pi}{8} - \frac{7\pi}{4} = \frac{15\pi}{8} - \frac{14\pi}{8} = \frac{\pi}{8}]

Plugging this back into the formula:

[d = \sqrt{64 + 25 - 80\cos\left(\frac{\pi}{8}\right)}]

Using the trigonometric identity:

[\cos\left(\frac{\pi}{8}\right) = \frac{\sqrt{2 + \sqrt{2}}}{2}]

Plugging this back into the formula:

[d = \sqrt{64 + 25 - 80*\frac{\sqrt{2 + \sqrt{2}}}{2}}]

[d = \sqrt{89 - 40\sqrt{2 + \sqrt{2}}}]

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

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