Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\theta $$

Short answer

The rectangular equation, in the form of parametric equations, are x = θ cos(θ) and y = θ sin(θ). The graph of the equation starts from the origin and spirals outwards, with the distance from the origin (r) always equal to the angle (θ).

Step-by-step solution

Converting the equation from polar to rectangular form

Use the relationships x = r cos(θ) and y = r sin(θ) to convert the polar equation r = θ to rectangular form. According to the polar equation, the radius or distance from the origin (r) is equal to the angle (θ). Substitute r = θ into both equations to convert the form: \[ x = θ \cos(θ) \] \[ y = θ \sin(θ) \]

Recognize the parametric form

The equations obtained in Step 1 form a parametric equation in θ. where x and y are represented as functions of θ. It has no simple equation in x and y, but some of its properties can be deduced from the polar equation.

Sketching the graph

To graph the equation in rectangular coordinates, note that for every value of the angle θ, the distance r from the origin is equal to the angle. The graph therefore will start from the origin and spiral outwards, because as θ increases, so does the radius.