Problem 21 - Essential Calculus

Question

Convert the rectangular equation to polar form and sketch its graph. $$ y=4 $$

Step-by-Step Solution

Answer

The polar form of the rectangular equation \(y = 4\) is \(r\sin(\theta) = 4\). The graph will be a circle with the centre at the origin and radius 4.

Step 1: Identify the equation in rectangular form

The given equation is \(y = 4\). This is clearly a horizontal line where y is always 4, regardless of the value of x.

Step 2: Convert the rectangular equation to polar form

For the polar coordinates, \((r, \theta)\), \(y = r\sin(\theta)\). Hence, for the given equation, we substitute y with r*sin(theta) to get \(r\sin(\theta) = 4\).

Step 3: Sketch its graph

The polar form of the equation, \(r\sin(\theta) = 4\), represents a line in polar coordinates. This line is the set of all points which are exactly 4 units from the origin with an angle \(\theta\) of the polar coordinates ranging all possible angles from 0 to \(2\pi\). Therefore, the graph will be a circle with the centre at the origin and radius 4.

Step 4: Final confirmation

Confirm that the polar equation and the graph correspond with the original rectangular equation \(y = 4\). It's clear that they do, because the polar equation simplifies to the rectangular equation if we use the relationship \(y = r\sin(\theta)\), and the graph shows a horizontal line 4 units above the origin.