# What is the distance between the following polar coordinates?: # (2,(5pi)/12), (1,(3pi)/12) #

The distance is

Use the Law of Cosines:

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To find the distance between two polar coordinates (r₁, θ₁) and (r₂, θ₂), you can use the formula:

[d = \sqrt{r₁^2 + r₂^2 - 2r₁r₂\cos(θ₂ - θ₁)}]

Plugging in the given values: [r₁ = 2, θ₁ = \frac{5\pi}{12}, r₂ = 1, θ₂ = \frac{3\pi}{12}]

[d = \sqrt{2^2 + 1^2 - 2(2)(1)\cos\left(\frac{3\pi}{12} - \frac{5\pi}{12}\right)}]

[d = \sqrt{4 + 1 - 4\cos\left(-\frac{\pi}{6}\right)}]

[d = \sqrt{5 - 4\cos\left(-\frac{\pi}{6}\right)}]

[d = \sqrt{5 - 4\left(\frac{\sqrt{3}}{2}\right)}]

[d = \sqrt{5 - 2\sqrt{3}}]

[d ≈ 1.05]

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