The rectangular coordinates (x , y) and polar coordinates (R , t) are related as follows.
y = R sin t and x = R cos t
R^{ 2} = x^{ 2} + y^{ 2} and tan t = y / x
To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant.
Angle t is in the range [0 , 2Pi) or [0 , 360 degrees).
Problem 1: Convert the polar coordinates (5 , 2.01) and (0.2 , 53 ^{ o}) to rectangular coordinates to three decimal places.
Solution to Problem 1:
 For the first point (5 , 2.01) R = 5 and t = 2.01 and is in radians. Set your calculator to radians and use the above formulas for x and y in terms of R and t to obtain:
x = R cos t = 5 cos 2.01 = 2.126
y = R sin t = 5 sin 2.01 = 4.525
 For the second point (0.2 , 53 ^{ o}) R = 0.2 and t = 53 ^{ o} and is in degrees. Set your calculator to degrees and use the above formulas for x and y in terms of R and t to obtain:
x = R cos t = 0.2 cos 53 = 0.120
y = R sin t = 0.2 sin 53 = 0.160
Problem 2: Convert the rectangular coordinates (1 , 1) and (2 ,4) to polar coordinates to three decimal places. Express the polar angle t in degrees and radians.
Solution to Problem 2:
 We first find R using the formula R = sqrt [x^{ 2} + y^{ 2}] for the point (1 , 1).
R = sqrt [x^{ 2} + y^{ 2}] = sqrt [1 + 1] = sqrt ( 2 )
 We now find tan t using the formula tan t = y / x.
tan t = 1 / 1
 Using the arctan function of the calculator, we obtain.
t = Pi / 4 or t = 45 ^{ o}
 Point (1 , 1) in rectangular coordinates may be written in polar for as follows.
( sqrt ( 2 ) , Pi / 4 ) or ( sqrt ( 2 ) , 45 ^{ o} )
 Let us find find R using for the point (2 , 4).
R = sqrt [x^{ 2} + y^{ 2}] = sqrt [4 + 16] = sqrt ( 20 ) = 2 sqrt ( 5 )
 We now find tan t.
tan t =  4 /  2 = 2
 Using the arctan function of the calculator, we obtain.
t = 1.107 or t = 63.435^{ o}
 BUT since the rectangular coordinates x and y are both negative, the point is in quadrant III and we need to add Pi or 180 ^{ o} to the value of t given by the calculator. Hence the polar angle t is given by
t = 4.249 or t = 243.435 ^{ o}
 Point (2 , 4) in rectangular coordinates may be written in polar for as follows.
( 5 sqrt ( 2 ) , 4.249 ) or ( 5 sqrt ( 2 ) , 243.435 ^{ o} )
More references on polar coordinates and trigonometry topics.
Polar Coordinates.
Trigonometry Tutorials and Problems.
