What is the distance between the following polar coordinates?: # (7,(-2pi)/3), (5,(-pi)/6) #

Question

What is the distance between the following polar coordinates?: #   (7,(-2pi)/3), (5,(-pi)/6)   #

Paisley Johnson · Accepted Answer

#sqrt74# units.

It's probably easiest to do this by first converting your polar coordinates into Cartesian coordinates—of course, polar coordinates are non-unique so theoretically they could translate to different Cartesian coordinates. We shall take the obvious set of points, however. To convert, remember that for a set of polar coordinates #[r, theta]#, #r=sqrt(x^2+y^2)#. Secondly, #x=r cos theta# and #y= r sin theta#. For the point #[7, -(2pi)/3]#, we get: #x=7cos(-(2pi)/3)# and #y=7sin(-(2pi)/3)#. #cos(-(2pi)/3) = cos((2pi)/3) = -cos(pi/3) = -1/2# #sin(-(2pi)/3) = -sin((2pi)/3) = -sin(pi/3) = -sqrt3/2# With a bit of further calculation: #x=-7/2# and #y=(-7sqrt3)/2# For the point #[5, -pi/6]#, we get: #x=5cos(-pi/6)# and #y=5sin(-pi/6)#. #cos(-pi/6) = cos(pi/6) = sqrt3/2#. #sin(-pi/6) = -sin(pi/6) = 1/2#. With a further bit of calculation: #x=(5sqrt3)/2# and #y=-5/2#. The next bit is the easiest; simply apply the formula for the distance between two points #D_"xy"=sqrt((x_2-x_1)^2+(y_2-y_1)^2)# with the two points #(-7/2,(-7sqrt3)/2)# and #((5sqrt3)/2, -5/2)#. The final answer after inserting those numbers is: #D_"xy"=sqrt74#

Ezekiel Alexander · Answer

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

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

Substitute the given values into the formula:

r₁ = 7, θ₁ = -2π/3
r₂ = 5, θ₂ = -π/6

Distance = √(7² + 5² - 2*7*5*cos((-π/6) - (-2π/3)))

Distance = √(49 + 25 - 70*cos(π/3))

Distance = √(49 + 25 - 70*(1/2))

Distance = √(49 + 25 - 35)

Distance = √39

So, the distance between the two polar coordinates is √39.