Convert Polar to Rectangular Coordinates and Vice Versa

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

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.