Question

Sketch a graph of the polar equation. $$ r=\frac{1}{\theta} $$

Short answer

The graph of \( r = \frac{1}{\theta} \) produces a spiral shape graph which spirals inward in a counter-clockwise direction as we increase from \(\theta = 0\).

Step-by-step solution

Identify the nature of the graph

The graph of this function yields a spiral because as the angle θ increases, the radius, \( r = \frac{1}{\theta} \), decreases reciprocally. This causes the points to get closer and closer to the origin as the angle increases.

Plotting the graph

Start plotting points. For simplicity, use multiples of \(\pi\) because these are easier to calculate. For example, at \(\theta = \pi\), \(r = \frac{1}{\pi}\). Plot this point in the corresponding direction of \(\theta = \pi\) at a distance of \(\frac{1}{\pi}\) units. Repeat this process with other multiples of \(\pi\) to obtain more points. Then, smoothly join these points to form the general shape of the spiral graph.

Indicate direction of rotation

To indicate the direction, start from \(\theta = 0\) (if \(\theta = 0\), r is undefined, thus the graph does not start exactly at the origin, but it spirals toward it as \(\theta\) increases). Then as \(\theta\) increases, r decreases ,thus spiraling inward counter-clockwise.