# What is the distance between the following polar coordinates?: # (3,(-7pi)/3), (1,(3pi)/4) #

The distance formula for polar coordinates is

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

Distance = √(r₁² + r₂² - 2 * r₁ * r₂ * cos(θ₂ - θ₁))

Where: r₁ and r₂ are the radii (distances from the origin) of the polar coordinates. θ₁ and θ₂ are the angles (in radians) of the polar coordinates.

Given the polar coordinates: (3, (-7π)/3) → r₁ = 3, θ₁ = (-7π)/3 (1, (3π)/4) → r₂ = 1, θ₂ = (3π)/4

Plug the values into the formula:

Distance = √(3² + 1² - 2 * 3 * 1 * cos((3π)/4 - (-7π)/3))

Calculate the cosine of the difference of the angles:

cos((3π)/4 - (-7π)/3) = cos(3π/4 + 7π/3) ≈ cos(13π/12)

Now, substitute back into the distance formula:

Distance ≈ √(9 + 1 - 6 * cos(13π/12))

Calculate the cosine value and simplify:

cos(13π/12) ≈ -√3/2

Distance ≈ √(9 + 1 - 6 * (-√3/2)) ≈ √(10 + 3√3) ≈ √(10 + 3√3)

So, the distance between the two polar coordinates is approximately √(10 + 3√3).

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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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