HOW TO CONVERT BETWEEN POLAR AND RECTANGULAR EQUATIONS

Convert each rectangular equation to polar form.

Example 1 :

x = 3

Solution :

x = 3

Substitute x = r cos θ.

r cos θ = 3

Divide both sides by cos θ. 

Example 2 :

x² + y² = 16

Solution :

x² + y² = 16

Substitute r² for x² + y².

r² = 16

Take square root on both sides.

r = ±4

Example 3 :

3x + 5y = 7

Solution :

3x + 5y = 7

Substitute r cos θ for x and r sin θ for y.

3r cos θ + 5r sin θ = 7

r(3 cos θ + 5 sin θ) = 7

Divide both sides by (3 cos θ + 5 sin θ).

Example 4 :

x² = 5y

Solution :

x² = 5y

Substitute r cos θ for x and r sin θ for y.

(r cos θ)² = 5(r sin θ)

r² cos² θ = 5r sin θ

Divide both sides by r cos² θ.

Example 5 :

x² + y² = 8y

Solution :

x² + y² = 8y

Substitute r² for x² + y² and r sin θ for y.

r² = 8r sin θ

Divide both sides by r.

r = 8 sin θ

Convert each polar equation to rectangular form.

Example 6 :

θ = 60°

Solution :

θ = 60°

Take tan on both sides.

tan θ = tan 60°

tan θ = √3

Example 7 :

r = -sec θ

Solution :

r = -sec θ

r cos θ = -1

Substitute x for r cos θ.

x = -1

Example 8 :

Solution :

Multiply both sides by (4cos θ + 6sin θ).

r(4cos θ + 6sin θ) = 5

4r cos θ + 6r sin θ = 5

Substitute x for r cos θ and y for r sin θ.

4x + 6y = 5

Example 9 :

Solution :

Multiply both sides by (1 - cos θ).

r(1 - cos θ) = 1

r - r cos θ = 1

x² + y² = x² + 2x + 1

Subtract x² from both sides.

y² = 2x + 1

Example 10 :

Solution :

Multiply both sides by (2 - sin θ).

r(2 - sin θ) = 6

2r - r sin θ = 6

4(x² + y²) = y² + 12y + 36

4x² + 4y² = y² + 12y + 36

4x² + 3y² - 12y - 36 = 0

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