# What is the distance between the following polar coordinates?: # (1,(pi)/4), (12,(9pi)/8) #

Distance

Distance between two polar coordinates formula

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

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

Where ( (r_1, \theta_1) ) and ( (r_2, \theta_2) ) are the polar coordinates.

Substituting the given values, we get:

[ d = \sqrt{1^2 + 12^2 - 2(1)(12)\cos\left(\frac{9\pi}{8} - \frac{\pi}{4}\right)} ]

[ d = \sqrt{1 + 144 - 24\cos\left(\frac{9\pi}{8} - \frac{\pi}{4}\right)} ]

[ d = \sqrt{145 - 24\cos\left(\frac{9\pi}{8} - \frac{\pi}{4}\right)} ]

[ d = \sqrt{145 - 24\cos\left(\frac{9\pi}{8} - \frac{2\pi}{8}\right)} ]

[ d = \sqrt{145 - 24\cos\left(\frac{\pi}{8}\right)} ]

[ d = \sqrt{145 - 24\left(\frac{\sqrt{2+\sqrt{2}}}{2}\right)} ]

[ d = \sqrt{145 - 12\sqrt{2+\sqrt{2}}} ]

[ d \approx \sqrt{145 - 12(2.613)} ]

[ d \approx \sqrt{145 - 31.356} ]

[ d \approx \sqrt{113.644} ]

[ d \approx 10.657 ] (rounded to three decimal places)

So, the distance between the polar coordinates ( (1,\frac{\pi}{4}) ) and ( (12,\frac{9\pi}{8}) ) is approximately 10.657 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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- What is the Cartesian form of #(12,(9pi)/3))#?

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