Transforming polar equations into rectangular coordinates

 Transforming polar equations into rectangular coordinates leads to find a relation between x and y. To do so, we use the formulas that allow to convert points between coordinates.

Example

Rewrite each of the following equations into rectangular coordinates and identify the graph.

a. θ = π/3

b.  r = 3

c. r = 6cosθ-8sinθ

Solution

a,Let's take the tangent of of both sides:

tanӨ = tan𝝅/3 = ⎷3

Let's substitute tanӨ by y/x:

y/x = ⎷3   y = x⎷3. This is the equation of a straight line passing through the origin and of slope ⎷3. In general any polar equation of the form  θ = K represents a straight line passing through the pole and with slope tanK.

b. Let's use the equation x² + y² = r². Let's substitute r by 3:  x² + y² = 9. This is the equation of a circle centered at the origin. In general, any polar equation of the form r = k where k is a positive constant represents a circle centered at the origin and with radius k.

c.  Let's multiply both sides by r:

r² = 6rcosθ-8rsinθ.

Let's substitute rcosθ by x and rsinθ by y.

r² = 6x - 8y.

Let's use the equation x² + y² = r² and substitute r² :

x² + y² = 6x - 8y

x² -  6x + y² + 8y = 0

x² -  6x +9-9 + y² + 8y + 16-16 = 0

(x² -  6x +9) +(y² + 8y + 16) -25= 0

(x-3)² + (y + 4)² = 25

This the equation of a circle centered at the point (3, -4) with radius r = 5.

Practice

Rewrite the equation r = secθtanθ in rectangular coordinates and identify its graph.