What is the distance between ##(1 ,( pi)/4 )## and ##(4 , ( pi )/2 )##?

##~~3.37##, I changed the points into Cartesian, then used the Pythagoras distance equation

The distance ##S## between two points on a plane is given by:

##S = sqrt(Delta x^2 + Delta y^2)## ##S = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)##

And using a conversion formula:

##x = r cos theta## ##y = r sin theta##

It proves more convenient to change our points to Cartesian before using the distance equation.

##(r,theta) -> (x,y)## ##(1,pi/4) -> (1/sqrt2,1/sqrt2)## ##(4,pi/2) -> (0,4)##

Choosing point 1 as ##(1/sqrt2,1/sqrt2)## and point 2 as ##(0,4)## we find:

##S = sqrt((0-1/sqrt2)^2 + (4-1/sqrt2)^2)##

##S = sqrt(1/2 + 16-8/sqrt2 +1/2)##

##S = sqrt(17-8/sqrt(2))##

At this point I gave up on getting a nice exact answer and used a calculator.

##S~~ 3.37##