What is the distance between the following polar coordinates?: # (5,(-5pi)/3), (5,(11pi)/6) #

Answer 1

#5sqrt2#

Before finding the distance between these points, it is necessary to convert them to rectangular coordinates. These points have been plotted as in the given figure and explained as follows:

To plot the point #(5, (-5pi)/3)#,go #(5pi)/3# radians or #300^o# clockwise from x- axis and mark point P at a radial distance of 5 units from the origin O. This is shown in the black pen. Now draw a perpendicular PR from point P to the x -axis. In the right triangle POR, #angle#POR is #pi/3#. OR=5 cos#pi/3= 5/2# and PR= 5 sin#pi/3= (5sqrt3) /2#. Thus rectangular coordinates of P are #(5/2, (5sqrt3)/2)#

To plot the point #(5, (11pi)/6)#, go #(11pi)/6# or #330^o# counterclockwise from x -axis and mark point Q at a radial distance of 5 units from the origin O. This is shown in red pen. Now draw a perpendicular QS from point Q to the x -axis. In the right triangle QOS, #angle# QOS is #(-pi)/6#. OS =#5 cos(-pi/6)= (5sqrt3)/2# and QS= 5sin#(-pi)/6 = -5/2# Thus rectangular coordinates of Q are #((5sqrt3)/2, -5/2)#

Distance between points P and Q= #sqrt( (5/2-(5sqrt3)/2)^2 +((5sqrt3)/2 +5/2)^2)#

=#sqrt(2(25/4 +75/4)#

= 5#sqrt2#

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Answer 2

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

Distance = sqrt(r1^2 + r2^2 - 2 * r1 * r2 * cos(θ2 - θ1))

Given: r1 = 5, θ1 = -5π/3 r2 = 5, θ2 = 11π/6

Substituting the values: Distance = sqrt(5^2 + 5^2 - 2 * 5 * 5 * cos((11π/6) - (-5π/3)))

Calculate cos((11π/6) - (-5π/3)): = cos(11π/6 + 5π/3) = cos(11π/6 + 10π/6) = cos(21π/6) = cos(7π/2) = 0

Substitute back into the formula: Distance = sqrt(5^2 + 5^2 - 2 * 5 * 5 * 0) = sqrt(25 + 25 - 0) = sqrt(50) = 5√2

Therefore, the distance between the given polar coordinates is 5√2.

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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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