Converting Polar Equations to Rectangular Equations

Key Formulas

• Right Triangle Relationships:
  • [x^2 + y^2 = r^2]
  • [x = r \cos(\theta)]
  • [y = r \sin(\theta)]
  • [\tan(\theta) = \frac{y}{x}]

Distinguishing Between Polar and Rectangular Equations

• Polar Equations: Involve variables ( r ) and ( \theta ).
  • Example: ( r = 5 \sin(\theta) )
  • Example: ( r = 7 )
  • Example: ( \theta = \frac{\pi}{4} )
• Rectangular Equations: Involve variables ( x ) and ( y ).
  • Example: ( x^2 + y^2 = 4 )
  • Example: ( x = 3 )
  • Example: ( x^2 = 4y )

Converting Examples

Example 1: ( r = 7 )

• Square both sides: ( r^2 = 49 )
• Use ( x^2 + y^2 = r^2 ):
  • ( x^2 + y^2 = 49 )

Example 2: ( r = 5 )

• Similar to Example 1:
  • ( x^2 + y^2 = 25 )

Example 3: ( \theta = \frac{\pi}{4} )

• Take the tangent of both sides: ( \tan(\theta) = 1 )
• Use ( \tan(\theta) = \frac{y}{x} ):
  • ( \frac{y}{x} = 1 ) ⟹ ( y = x )

Example 4: ( \theta = 0 )

• Take the tangent of both sides: ( \tan(0) = 0 )
• Use ( \frac{y}{x} = 0 ):
  • ( y = 0 )

Example 5: ( \theta = \frac{\pi}{2} )

• ( \tan(\theta) ) is undefined
• For ( \frac{y}{x} ) to be undefined, ( x = 0 )
• Answer: ( x = 0 )

Example 6: ( r \sin(\theta) = 5 )

• Use ( y = r \sin(\theta) ):
  • ( y = 5 )
• Similarly, ( r \cos(\theta) = 4 ) ⟹ ( x = 4 )

Example 7: ( r = 3 \csc(\theta) )

• ( \csc(\theta) = \frac{1}{\sin(\theta)} )
• Multiply both sides by ( \sin(\theta) ):
  • ( r \sin(\theta) = 3 )
  • ( y = 3 )

Example 8: ( r = 4 \sec(\theta) )

• ( \sec(\theta) = \frac{1}{\cos(\theta)} )
• Multiply both sides by ( \cos(\theta) ):
  • ( r \cos(\theta) = 4 )
  • ( x = 4 )

Example 9: ( r = 3 \sin(\theta) )

• Multiply both sides by ( r ):
  • ( r^2 = 3r \sin(\theta) )
  • ( x^2 + y^2 = 3 y )

Example 10: ( r = 4 \cos(\theta) )

• Multiply both sides by ( r ):
  • ( r^2 = 4r \cos(\theta) )
  • ( x^2 + y^2 = 4x )

Example 11: ( r = 3 \cos(\theta) + 5 \sin(\theta) )

• Multiply both sides by ( r ):
  • ( r^2 = 3r \cos(\theta) + 5r \sin(\theta) )
  • ( x^2 + y^2 = 3x + 5y )

Example 12: ( r = \frac{5}{2 \cos(\theta) + 3 \sin(\theta)} )

• Multiply both sides by ( 2 \cos(\theta) + 3 \sin(\theta) ):
  • ( r(2 \cos(\theta) + 3 \sin(\theta)) = 5 )
  • ( 2x + 3y = 5 )

Example 13: ( r^2 \sin(2 \theta) = 8 )

• Use double angle formula: ( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) )
• Divide by 2:
  • ( r^2 \sin(\theta) \cos(\theta) = 4 )
  • ( y \cdot x = 4 ) ⟹ ( y = \frac{4}{x} )

Example 14: ( r = \frac{8}{\cos(\theta)} )

• Multiply both sides by ( \cos(\theta) ):
  • ( r \cos(\theta) = 8 )
  • ( x = 8 )

Example 15: ( r = \frac{5 \cos(\theta)}{\sin^2(\theta)} )

• Multiply both sides by ( \sin(\theta) ):
  • ( r \sin(\theta) = \frac{5 \cos(\theta)}{\sin(\theta)} )
  • ( y = 5 \cot(\theta) )
  • ( y^2 = 5x )

Example 16: ( r = \sin(\theta) \cos^2(\theta) )

• Multiply both sides by ( r^3 ):
  • ( r^4 = r^3 \sin(\theta) \cos^2(\theta) )
  • ( (x^2 + y^2)^2 = x y x ) ⟹ ( x^2 + y^2 = x \sqrt{y} )