Worksheet: Conversion between Rectangular and Polar Equations

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

Q1:

Consider the polar equation 𝑟 = 2 𝜃 c o s . 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 𝑟 .

  • A 𝑟 = 𝑟 𝜃 2 c o s
  • B 𝑟 = 2 𝜃 2 c o s
  • C 𝑟 = 2 𝑟 𝜃 c o s
  • D 𝑟 = 2 𝑟 𝜃 2 c o s
  • E 𝑟 = 𝑟 𝜃 c o s

Use the fact that 𝑥 = 𝑟 𝜃 c o s to simplify the expression.

  • A 𝑟 = 2 𝑥 2
  • B 𝑟 = 𝑥
  • C 𝑟 = 2 𝑥
  • D 𝑟 = 𝑥 2
  • E 2 𝑟 = 𝑥 2

Given that 𝑥 = 𝑟 𝜃 c o s and 𝑦 = 𝑟 𝜃 s i n , we can use the Pythagorean theorem to show that 𝑥 + 𝑦 = 𝑟 2 2 2 . Use this to eliminate the 𝑟 2 in the previous expression.

  • A 𝑥 + 𝑦 = 𝑥 2 2 2
  • B 𝑥 + 𝑦 = 𝑥 2 2
  • C 𝑥 + 𝑦 = 2 𝑥 2 2
  • D 𝑥 + 𝑦 = 4 𝑥 2 2 2
  • E 𝑥 + 𝑦 = 𝑥 2 2 2

Q2:

Convert 𝑟 = 2 𝜃 s e c into cartesian form.

  • A 𝑥 = 2 2
  • B 𝑦 = 2
  • C 𝑦 = 2 2
  • D 𝑥 = 2
  • E 𝑥 = 4

Q3:

Consider the Cartesian equation 𝑦 = 2 𝑥 + 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 𝑥 = 𝑟 𝜃 c o s to eliminate 𝑥 .

  • A 𝑦 = 𝑟 𝜃 + 3 c o s
  • B 𝑦 = 2 ( 𝑟 𝜃 + 3 ) c o s
  • C 𝑦 = 2 𝑟 𝜃 c o s
  • D 𝑦 = 2 𝑟 𝜃 + 3 c o s
  • E 𝑦 = 2 𝑟 𝜃 3 c o s

Now, use the fact that 𝑦 = 𝑟 𝜃 s i n to eliminate 𝑦 .

  • A 𝑟 𝜃 = 2 𝑟 𝜃 + 3 s i n c o s
  • B 𝑟 𝜃 = 2 𝑟 𝜃 s i n c o s
  • C 𝑟 𝜃 = 2 ( 𝑟 𝜃 + 3 ) s i n c o s
  • D 𝑟 𝜃 = 𝑟 𝜃 + 3 s i n c o s
  • E 𝑟 𝜃 = 2 𝑟 𝜃 3 s i n c o s

Finally, make 𝑟 the subject.

  • A 𝑟 = 3 𝜃 2 𝜃 s i n c o s
  • B 𝑟 = 3 𝜃 + 2 𝜃 s i n c o s
  • C 𝑟 = 3 𝜃 2 𝜃 s i n c o s
  • D 𝑟 = 3 𝜃 𝜃 s i n c o s
  • E 𝑟 = 3 𝜃 + 𝜃 s i n c o s

Q4:

Convert the equation 𝑥 + 𝑦 = 2 5 2 2 into polar form.

  • A 𝑟 = 2 5 2
  • B 𝑟 = 2 5
  • C 𝑟 = 5 0
  • D 𝑟 = 5
  • E 𝑟 = 6 2 5

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