What is the distance between #(-6 , pi/2 )# and #(5, pi/6 )#?

Answer 1

#\sqrt{91}# units.

Since the coordinates are polar coordinates, we imagine a triangle #OAB# with #O# at the origin, #A# at #(-6,\pi/2)# and #B# at #(5,\pi/6)#. We will use the Law of Cosines to get the length of #AB#.

First convert side #OA# to a positive length by writing its coordinates as #(+6,-\pi/2)#, changing the sign of the radius and compensating by subtracting #\pi# from the angle.

So #OA=6# and #OB=5#. Next we need angle #O# which is the difference between the angular coordinates after we have made the radial coordinates positive (see above). Thus

#\pi/2-(-\pi/6)=2\pi/3#.

Now apply the Law of Cosines:

#(AB)^2=(OA)^2+(OB)^2-2(OA)(OB)\cos(angle O)#

#=6^2+5^2-2(6)(5) \cos(2\pi/3)#

As #\cos(2\pi/3)=-1/2# this gives #(AB)^2=91#.

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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.