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

Answer 1

The distance formula for polar coordinates is

#d=sqrt(r_1^2+r_2^2-2r_1r_2Cos(theta_1-theta_2)# Where #d# is the distance between the two points, #r_1#, and #theta_1# are the polar coordinates of one point and #r_2# and #theta_2# are the polar coordinates of another point. Let #(r_1,theta_1)# represent #(3,(-7pi)/3)# and #(r_2,theta_2)# represent #(1,(3pi)/4)#. #implies d=sqrt(3^2+1^2-2*3*1Cos((-7pi)/3-(3pi)/4)# #implies d=sqrt(9+1-6Cos((-28pi-9pi)/12)# #implies d=sqrt(10-6Cos((-37pi)/12)# #implies d=sqrt(10-6(-0.9659))# #implies d=sqrt(10+5.7954)=sqrt(15.7954)=3.9743# units #implies d=3.9743# units (approx) Hence the distance between the given points is #3.9743# units.
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Answer 2

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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Answer from HIX Tutor

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.

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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