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

Examples on Converting Polar and Rectangular Coordinates

Example 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°), R = 0.2 and t = 53° 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

Example 2

• We first find R using the formula R = √ [x ² + y ²] for the point (1 , 1).
  R = √ [x ² + y ²] = √ [1 + 1] = √ ( 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.
  ( √ ( 2 ) , π / 4 ) or ( √ ( 2 ) , 45 ^o )
  
  
• Let us find find R using for the point (-2 , -4).
  R = √ [x ² + y ²] = √ [4 + 16] = √ ( 20 ) = 2 √ ( 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 π or 180° to the value of t given by the calculator. Hence the polar angle t is given by
  t = 4.249 or t = 243.435°
  
• Point (-2 , -4) in rectangular coordinates may be written in polar for as follows.
  ( 2 √ ( 5 ) , 4.249 ) or ( 2 √ ( 5 ) , 243.435° )
  
  

More References and Links to Polar Coordinates and Trigonometry