Question

In Exercises \(15-20,\) find an equation in rectangular coordinates for the equation given in cylindrical coordinates, and sketch its graph. $$ \theta=\pi / 6 $$

Short answer

The equation in rectangular coordinates is a straight line that makes a 30-degree angle with the positive x-axis.

Step-by-step solution

Start by understanding the relationship between cylindrical and rectangular coordinates

In cylindrical coordinates, the angle \(\theta\) is the angle made with the positive x-axis. The given equation \(\theta=\pi / 6\) means that the angle each point makes with the positive x-axis is \(\pi / 6\). So all the points must lie on a line which makes an angle of \(\pi / 6\) with the positive x-axis.

Convert the value of \(\theta=\pi / 6\) into degrees

The value of \(\pi / 6\) is equal to 30 degrees. So what we are trying to find are all the points in the plane that form an angle of 30 degrees with the positive x-axis.

Sketching the graph

On the graph, mark the angle of 30 degrees starting from the positive x-axis and draw a line. The line formed will represent the graph of the equation \(\theta=\pi / 6\). Please remember that the direction is determined by rotating anti-clockwise from the x-axis