# What is the distance between the following polar coordinates?: # (3,(-4pi)/3), (4,(-5pi)/6) #

First, write the polar coordinates in cartesian form. You will need the parametric form to help you with that:

Plug in for the first point:

Plug in for the second point:

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

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

Using the given polar coordinates: r₁ = 3, θ₁ = -4π/3 r₂ = 4, θ₂ = -5π/6

Plugging these values into the formula:

Distance = √(3² + 4² - 2 * 3 * 4 * cos((-5π/6) - (-4π/3)))

Calculating the cosine of the difference of the angles and substituting back:

cos((-5π/6) - (-4π/3)) = cos(π/2) = 0

Substituting back into the distance formula:

Distance = √(9 + 16 - 2 * 3 * 4 * 0)

Distance = √(9 + 16 - 0)

Distance = √25

Distance = 5

So, the distance between the given polar coordinates is 5 units.

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

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

Using the given coordinates:

[ r_1 = 3, \theta_1 = -\frac{4\pi}{3}, ] [ r_2 = 4, \theta_2 = -\frac{5\pi}{6} ]

[ \text{Distance} = \sqrt{3^2 + 4^2 - 2(3)(4)\cos\left(-\frac{5\pi}{6} - \left(-\frac{4\pi}{3}\right)\right)} ]

[ \text{Distance} = \sqrt{9 + 16 - 24\cos\left(-\frac{5\pi}{6} + \frac{4\pi}{3}\right)} ]

[ \text{Distance} = \sqrt{25 - 24\cos\left(\frac{\pi}{2}\right)} ]

[ \text{Distance} = \sqrt{25 - 24(0)} ]

[ \text{Distance} = \sqrt{25} ]

[ \text{Distance} = 5 ]

So, the distance between the polar coordinates (3,(-4π)/3) and (4,(-5π)/6) is 5 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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