Lesson Worksheet: Conversion between Rectangular and Polar Equations Mathematics • Higher Education

In this worksheet, we will practice converting equations from polar to rectangular form and vice versa.


Consider the polar equation π‘Ÿ=2πœƒcos. Complete the following steps to help you find the Cartesian form of the equation by writing an equivalent equation each time.

Multiply both sides of the equation through by π‘Ÿ.

  • Aπ‘Ÿ=π‘Ÿπœƒcos
  • Bπ‘Ÿ=2π‘Ÿπœƒcos
  • Cπ‘Ÿ=2π‘ŸπœƒοŠ¨cos
  • Dπ‘Ÿ=2πœƒοŠ¨cos
  • Eπ‘Ÿ=π‘ŸπœƒοŠ¨cos

Use the fact that π‘₯=π‘Ÿπœƒcos to simplify the expression.

  • Aπ‘Ÿ=2π‘₯
  • Bπ‘Ÿ=π‘₯
  • Cπ‘Ÿ=π‘₯
  • D2π‘Ÿ=π‘₯
  • Eπ‘Ÿ=2π‘₯

Given that π‘₯=π‘Ÿπœƒcos and 𝑦=π‘Ÿπœƒsin, we can use the Pythagorean theorem to show that π‘₯+𝑦=π‘ŸοŠ¨οŠ¨οŠ¨. Use this to eliminate the π‘ŸοŠ¨ in the previous expression.

  • Aπ‘₯+𝑦=2π‘₯
  • Bπ‘₯+𝑦=π‘₯
  • Cπ‘₯+𝑦=π‘₯
  • Dπ‘₯+𝑦=4π‘₯
  • Eπ‘₯+𝑦=π‘₯2


Convert π‘Ÿ=2πœƒsec into cartesian form.

  • A𝑦=2
  • Bπ‘₯=2
  • Cπ‘₯=4
  • Dπ‘₯=2
  • E𝑦=2


Consider the Cartesian equation 𝑦=2π‘₯+3.Complete the following steps to find the polar form of the equation by writing an equivalent equation each time.

First, use the fact that π‘₯=π‘Ÿπœƒcos to eliminate π‘₯.

  • A𝑦=π‘Ÿπœƒ+3cos
  • B𝑦=2(π‘Ÿπœƒ+3)cos
  • C𝑦=2π‘Ÿπœƒcos
  • D𝑦=2π‘Ÿπœƒβˆ’3cos
  • E𝑦=2π‘Ÿπœƒ+3cos

Now, use the fact that 𝑦=π‘Ÿπœƒsin to eliminate 𝑦.

  • Aπ‘Ÿπœƒ=2(π‘Ÿπœƒ+3)sincos
  • Bπ‘Ÿπœƒ=2π‘Ÿπœƒsincos
  • Cπ‘Ÿπœƒ=2π‘Ÿπœƒβˆ’3sincos
  • Dπ‘Ÿπœƒ=π‘Ÿπœƒ+3sincos
  • Eπ‘Ÿπœƒ=2π‘Ÿπœƒ+3sincos

Finally, make π‘Ÿ the subject.

  • Aπ‘Ÿ=3πœƒβˆ’πœƒsincos
  • Bπ‘Ÿ=3πœƒ+πœƒsincos
  • Cπ‘Ÿ=3πœƒ+2πœƒsincos
  • Dπ‘Ÿ=βˆ’3πœƒβˆ’2πœƒsincos
  • Eπ‘Ÿ=3πœƒβˆ’2πœƒsincos


Convert the equation π‘₯+𝑦=25 into polar form.

  • Aπ‘Ÿ=5
  • Bπ‘Ÿ=50
  • Cπ‘Ÿ=625
  • Dπ‘Ÿ=25
  • Eπ‘Ÿ=252


Convert the rectangular equation 𝑦=4 to the polar form.

  • Aπ‘Ÿ=2
  • Bπ‘Ÿ=4πœƒsec
  • Cπ‘Ÿ=4πœƒcsc
  • Dπ‘Ÿ=4
  • Eπ‘Ÿ=2πœƒsec


Convert the rectangular equation π‘₯+𝑦=25 to the polar form.

  • Aπ‘Ÿ=5
  • Bπ‘Ÿ=5
  • Cπ‘Ÿ=√5
  • Dπ‘Ÿ=25


Convert the polar equation πœƒ=πœ‹4 to the rectangular form.

  • A𝑦=βˆ’2√2π‘₯
  • B𝑦=√22π‘₯
  • C𝑦=βˆ’π‘₯
  • D𝑦=βˆ’βˆš22π‘₯
  • E𝑦=π‘₯


Convert the polar equation π‘Ÿ=2 to the rectangular form.

  • Aπ‘₯+𝑦=2
  • Bπ‘₯+𝑦=4
  • Cπ‘₯βˆ’π‘¦=4
  • Dπ‘₯βˆ’π‘¦=2
  • Eπ‘₯+𝑦=2


Convert the polar equation π‘Ÿ=4πœƒβˆ’6πœƒcossin to the rectangular form.

  • A(π‘₯βˆ’2)+(𝑦+3)=13
  • B(π‘₯βˆ’2)βˆ’(𝑦+3)=√13
  • C(π‘₯βˆ’2)βˆ’(𝑦+3)=13
  • D(π‘₯+2)+(π‘¦βˆ’3)=13
  • E(π‘₯βˆ’2)+(𝑦+3)=√13


Consider the rectangular equation π‘₯βˆ’π‘¦=25.

Convert the given equation to the polar form.

  • Aπ‘Ÿ=25
  • Bπ‘Ÿ=√5
  • Cπ‘Ÿ=252πœƒοŠ¨csc
  • Dπ‘Ÿ=25
  • Eπ‘Ÿ=252πœƒοŠ¨sec

Which of the following is the sketch of the equation?

  • A
  • B
  • C
  • D
  • E

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