# 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#