What is the distance between #(2 , (5 pi)/8 )# and #(3 , (1 pi )/3 )#?
1 Answer
The distance between those two coordinates is
Explanation:
You can use the law of cosines to do that.
Let me illustrate why:
Polar coordinates
#(r, theta)# are defined by the radius#r# and the angle#theta# .Imagine lines leading from the pole to your respective polar coordinates. Those lines represent two sides of a triangle with lengths
#A = 3# and#B = 2# . The distance between those two coordinates being the third side,#C# .
Furthermore, the angle between
#A# and#B# can be computed as the difference between the two angles of your polar coordinates:
#gamma = (5pi)/8 - pi/3 = (7pi)/24#
Thus, the length of the side
#C^2 = A^2 + B^2 - 2AB cos(gamma)#
#= 3^2 + 2^2 - 2 * 3 * 2 * cos((7pi)/24)#
#= 13 - 12 cos((7pi)/24)#
#=> C = sqrt(13 - 12 cos((7pi)/24)) ~~2.39#