# What is the distance between the following polar coordinates?: # (5,(7pi)/4), (9,(11pi)/8) #

Distance between tow points knowing the polar coordinates is given by the formula using cosine rule

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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)}]

Substitute the given values:

[r_1 = 5, \theta_1 = \frac{7\pi}{4}] [r_2 = 9, \theta_2 = \frac{11\pi}{8}]

[d = \sqrt{5^2 + 9^2 - 2(5)(9)\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}]

[d = \sqrt{25 + 81 - 90\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}]

[d = \sqrt{106 - 90\cos\left(\frac{11\pi}{8} - \frac{7\pi}{4}\right)}]

[d = \sqrt{106 - 90\cos\left(\frac{\pi}{8}\right)}]

[d = \sqrt{106 - 90\cos\left(\frac{\pi}{8}\right)}]

[d \approx \sqrt{106 - 90(0.9239)}]

[d \approx \sqrt{106 - 83.55}]

[d \approx \sqrt{22.45}]

[d \approx 4.74]

So, the distance between the given polar coordinates is approximately (4.74).

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