Convert Equation from Polar to Rectangular Form

Equations in polar form are converted into rectangular form, using the relationship between polar and rectangular coordinates. Problems with detailed solutions are presented.

Problem 1: Convert the polar equation

Solution to Problem 1:

• We multiply both sides by R
  
  R = 4 sin t
  
  R ² = 4 R sin t
  
  
• We now use the relationship between polar and rectangular coordinates: R ² = x ² + y ² and y = R sin t to rewrite the equation as follows:
  
  x ² + y ² = 4 y
  
  x ² + y ² - 4 y = 0
  
  
• It is the equation of a circle.

Problem 2: Convert the polar equation

Solution to Problem 2:

• Expand the left side of the given equation.
  
  R(-2 sin t + 3 cos t) = 2
  
  -2 R sin t + 3 R cos t = 2
  
  
• Use y = R sin t and x = R cos t into the given equation to rewrite as follows:
  
  -2 y + 3 x = 2
  
  
• The above is the equation of a line.

Problem 3: Convert the polar equation

Solution to Problem 3:

• Rewrite the given equation as follows:
  
  t + Pi / 4 = 0
  
  t = - Pi / 4
  
  
• Take the tangent of both sides:
  
  tan t = -1
  
  
• Use y = R sin t and x = R cos t to write:
  
  tan t = sin t / cos t = R sin t / R cos t = y / x
  
  
• Hence:
  
  y / x = -1
  
  y = - x
  
  
• The above is the equation of a line.

More references on polar coordinates and trigonometry topics.