Problem 76 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r^{2}=4 \sin \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \(r^{2}=4 \sin \theta\) is a semicircle with radius 2 resting against the positive side of the y-axis in the polar coordinate system.

Step 1: Simplify the Equation

The equation given in the problem is in polar form. For easier manipulation, it’s helpful to simplify it into a more recognizable form. One way to do so is by taking the square root of each side. This results in the following equation: \( r = 2 \sqrt{\sin \theta} \).

Step 2: Identify the Graph's Features

The equation \( r = 2 \sqrt{\sin \theta} \) is recognizable as a semicircle in a rectangular coordinate system with a radius of 2. However, it should be noted that when \(\sin \theta\) is negative, \(\sqrt{\sin \theta}\) is undefined. Therefore, this graph will only exist when \(0 \leq \theta \leq \pi\). It's centered at the pole with the bottom touching the pole.

Step 3: Sketch the Graph

The graph will appear as a semicircle against the positive side of the y-axis, with the pole serving as the center. Starting from \(\theta = 0\), the circle will start from the pole and then, increase in radius until \(\theta = \frac{\pi}{2}\). After that, the radius will start to decrease again until \(\theta = \pi\), at which point the radius will be zero again. The sketch of the graph should reflect these properties.