Problem 31 - Essential Calculus

Question

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

Step-by-Step Solution

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 1: 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: \[ x = θ \cos(θ) \] \[ y = θ \sin(θ) \]

Step 2: 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.

Step 3: 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.