Problem 176 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The graph is a circle with symmetry about the line \( \theta = \frac{\pi}{2} \).

Step 1: Express in polar form

The given equation is already in polar form: \( r^2 = 4 \sin \theta \). In polar coordinates, it's often helpful to express \( r \) in terms of \( \theta \). So take the square root on both sides: \[r = \sqrt{4 \sin \theta} = 2 \sqrt{\sin \theta}.\]

Step 2: Identify possible values for \( \theta \)

Since the square root function is only defined for non-negative real numbers, we need \( \sin \theta \geq 0 \). This implies that \( \theta \) is in the range where the sine function is non-negative, namely:\[0 \leq \theta \leq \pi.\]

Step 3: Find symmetry in the graph

To determine symmetries, check substitution changes. Since \( r^2 = 4 \sin \theta \) is the same as itself when replacing \( \theta \) with \( \pi - \theta \), it shows symmetry with respect to the line \( \theta = \frac{\pi}{2} \). It does not show full rotational or reflective symmetry about the polar axis.

Step 4: Determine key points

Evaluate the equation for key values of \( \theta \):- For \( \theta = 0 \), \( r = 0 \).- For \( \theta = \frac{\pi}{2} \), \( r = 2 \) (since \( \sin \frac{\pi}{2} = 1 \)).- For \( \theta = \pi \), \( r = 0 \) (since \( \sin \pi = 0 \)).Plot these points to get a sense of the curve's shape.

Step 5: Sketch the curve

The plotted points form a semicircle in the first quarter of the plane (up to \( \theta = \frac{\pi}{2} \)), and a mirrored semicircle in the second quarter (due to the symmetry identified in Step 3), completing a circle as \( \theta \) progresses from 0 to \( \pi \). The full graph is like a complete top half-circle above the polar axis.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that represent curves in the polar coordinate system. In a polar coordinate system, each point is determined by a distance from a reference point (often called the origin) and an angle from a reference direction (usually the positive x-axis in the Cartesian system). This system is useful in certain applications where the relationship between the radius and the angle can be described more naturally.

A typical polar equation will have the form \( r = f(\theta) \), where \( r \) is the radial distance and \( \theta \) is the angle. For example, in the original exercise, the polar equation is \( r^2 = 4\sin\theta \). This tells us that the radial distance squared is four times the sine of the angle. Notice that this equation is already expressed in polar coordinates, making it convenient for graphing and analysis.

Transforming such equations can sometimes improve understanding or simplify graphing. For instance, in the solution, we expressed \( r = 2\sqrt{\sin \theta} \) by taking the square root of both sides. This transformation helps us identify the values at which \( r \) is defined, namely where \( \sin \theta \geq 0 \). Such polar transformations are essential for sketching the graph of the equation accurately.

Symmetry in Polar Graphs

Understanding symmetry in polar graphs can simplify graphing and analyzing complex curves. Symmetry helps in predicting the shape and behavior of curves by reducing the amount of calculation needed.

Two common types of symmetry to look for in polar graphs are:

• Symmetry with Respect to the Polar Axis: If the substitution \( (r, \theta) \) is equivalent to \( (r, -\theta) \), the graph is symmetric about the polar (horizontal) axis.
• Symmetry with Respect to the Line \( \theta = \frac{\pi}{2} \): If the substitution \( (r, \theta) \) is equivalent to \( (r, \pi - \theta) \), the graph is symmetric about the vertical line \( \theta = \frac{\pi}{2} \).

In the exercise provided, the polar equation \( r^2 = 4 \sin \theta \) showed symmetry with respect to \( \theta = \frac{\pi}{2} \). This means that the graph reflects evenly over this line, creating a mirrored image in the polar plot. Such progressive symmetry reduces the effort required to plot the full graph, as understanding the symmetrical properties can often help replicate half of the graph over the line or axis of symmetry.

Graphing Polar Curves

Graphing polar curves involves plotting points in the polar coordinate plane and recognizing the pattern that these points create. This method involves calculating \( r \) for several values of \( \theta \) and sketching their positions.

Let's go through the steps of graphing based on the original exercise:

• **Select key angles**: Determine where the function changes significantly, such as important angles like 0, \( \frac{\pi}{2} \), and \( \pi \).
• **Calculate corresponding \( r \)**: For each angle, use the polar equation to calculate \( r \). As mentioned in the solution, \( \theta = 0 \) gives \( r = 0 \); \( \theta = \frac{\pi}{2} \) gives \( r = 2 \); and \( \theta = \pi \) returns \( r = 0 \).
• **Draw the curve**: Map out the points in the polar plane and sketch the curve that fits these points. From \( \theta = 0 \) to \( \pi \), the points form a semicircle, which is mirrored due to the symmetry discussed earlier, forming a full circle in the upper half of the plane.

Creating the graph visually helps interpret the behavior of the equation. It's also valuable in understanding the spatial relations of different points as \( \theta \) varies. By effectively using polar equations and symmetry principles, you can efficiently represent even complex polar curves.