# What is the distance between the following polar coordinates?: # (2,(13pi)/4), (7,(-3pi)/8) #

So, as far as the position is concerned, the first point P is (1, -3/4pi).

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To find the distance between the polar coordinates ((2, \frac{13\pi}{4})) and ((7, \frac{-3\pi}{8})), you can use the formula:

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

Substituting the given values:

[ r_1 = 2, \quad r_2 = 7, \quad \theta_1 = \frac{13\pi}{4}, \quad \theta_2 = \frac{-3\pi}{8} ]

[ d = \sqrt{2^2 + 7^2 - 2(2)(7)\cos\left(\frac{-3\pi}{8} - \frac{13\pi}{4}\right)} ]

[ d = \sqrt{4 + 49 - 28\cos\left(\frac{-3\pi}{8} - \frac{13\pi}{4}\right)} ]

[ d = \sqrt{53 - 28\cos\left(\frac{-3\pi}{8} - \frac{13\pi}{4}\right)} ]

Now, find the difference of the angles inside the cosine function:

[ \frac{-3\pi}{8} - \frac{13\pi}{4} = \frac{-3\pi}{8} - \frac{26\pi}{8} = \frac{-29\pi}{8} ]

[ d = \sqrt{53 - 28\cos\left(\frac{-29\pi}{8}\right)} ]

[ d = \sqrt{53 - 28\cos\left(-\frac{\pi}{8}\right)} ]

[ d = \sqrt{53 - 28\cos\left(\frac{\pi}{8}\right)} ]

[ d = \sqrt{53 - 28\cos\left(\frac{\pi}{8}\right)} ]

[ d = \sqrt{53 - 28\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)} ]

[ d = \sqrt{53 - 14\sqrt{2+\sqrt{2}}} ]

This is the exact distance between the two polar coordinates. If you need a numerical approximation, you can calculate it using a calculator.

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- What is the Cartesian form of #(-49,(-3pi)/4)#?
- What is the area inside the polar curve #r=1#, but outside the polar curve #r=2costheta#?
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- What is the equation of the tangent line of #r=cos(2theta-pi/4)/sintheta - sin(theta-pi/8)# at #theta=(-3pi)/8#?
- What is the slope of the tangent line of #r=3sin(theta/2-pi/4)# at #theta=(3pi)/8#?

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