Question

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. $$ r=4 \sin \left(\frac{\theta}{3}\right) $$

Short answer

The polar curve exhibits symmetry about the origin and has loop patterns extending over \( 6\pi \) in \( \theta \).

Step-by-step solution

Understanding Polar Equations

A polar equation is given in the form \( r = f(\theta) \), where \( r \) is the radius and \( \theta \) is the angle. The equation \( r = 4 \sin\left(\frac{\theta}{3}\right) \) describes a function in polar coordinates.

Identify Symmetry

To determine symmetry, we check for symmetry about the polar axis, the line \( \theta = \frac{\pi}{2} \), and the pole. If \( f(\theta) = -f(-\theta) \), then the graph is symmetric about the origin (the pole). If \( r(-\theta) = r(\theta) \), the graph is symmetric about the polar axis. For \( r = 4 \sin\left(\frac{\theta}{3}\right) \), substituting \( -\theta \) gives \( r = 4 \sin\left(-\frac{\theta}{3}\right) = -4 \sin\left(\frac{\theta}{3}\right) \), showing symmetry about the origin.

Consider Domain of \( \theta \)

Given \( r = 4 \sin\left(\frac{\theta}{3}\right) \), note the sine function has a period of \( 2\pi \). Therefore, \( \frac{\theta}{3} \) means the function completes a full cycle for \( \theta \) ranging from 0 to \( 6\pi \). This should be considered for sketching the complete curve.

Calculating Key Points

To sketch the curve, evaluate \( r \) at several key angles. If \( \theta = 0, \pi, 2\pi, 3\pi, 4\pi, 5\pi, 6\pi \), the sine will be zero rendering \( r = 0 \). For \( \theta = \pi/3, 2\pi/3, 4\pi/3, 5\pi/3 \), \( r \) will be positive, indicating points at specific radii.

Sketching the Curve

Use the calculated points to sketch the curve. Since the sine function is periodic and changes between positive and negative values as the angle progresses, the curve forms loops that encircle the origin multiple times. For each \( \theta \) completing \( 2\pi \), you see a pattern from the key points form a petal shape.

Key concepts

Polar Coordinates

Polar coordinates are a way of plotting points on a plane using distance and angle. Imagine you are locating a point not by its position on a grid, but by how far away it is from a central point (the pole, similar to the origin in Cartesian coordinates) and at what angle. This is very useful, especially when dealing with curves and shapes that have circular symmetry, like spirals or roses.

In the given polar equation, \( r = 4 \sin\left(\frac{\theta}{3}\right) \), \( r \) represents the radial distance from the pole, and \( \theta \) is the angle from the positive x-axis (polar axis). Polar equations make it easier to express shapes that are difficult to describe with Cartesian coordinates, offering a different perspective for curve sketching.

The format \( r = f(\theta) \) indicates that for each value of \( \theta \), you have a corresponding radius \( r \), allowing you to sketch complex and beautiful curves like spirals, circles, and more.

Symmetry in Graphs

Symmetry in graphs of polar equations reveals how the graph may reflect across certain lines or points. This can simplify the process of sketching the graph by predicting its shape.

Symmetry can occur in the following ways:

• Pole Symmetry: If the equation is symmetric about the origin, substituting \(-\theta\) into the equation gives \(-r\). In the example of \( r = 4 \sin\left(\frac{\theta}{3}\right) \), substituting \(-\theta\) results in a negative \( r \), indicating a reflection through the pole.
• Polar Axis Symmetry: The graph is symmetric about the polar axis (like the x-axis in Cartesian coordinates) when \( f(\theta) = f(-\theta) \).
• Line \( \theta = \frac{\pi}{2} \) Symmetry: Checking \( f\left(\frac{\pi}{2} - \theta\right) \) may reveal symmetry about the vertical line equivalent to \( y \)-axis symmetry.

Recognizing these symmetries helps plot the curves more efficiently, knowing portions of the curve simply mirror across an axis or point.

Sine Function

The sine function, a fundamental trigonometric function, oscillates between -1 and 1, creating wave-like patterns on graphs. Understanding these oscillations is crucial when sketching curves, particularly in polar coordinates where the sine function is often present as the core of the radial equation.

For the equation \( r = 4 \sin\left(\frac{\theta}{3}\right) \), the sine function's periodic behavior governs the shape and occurrence of the curve's looping pattern:

• Amplitude: The amplitude here is 4, meaning the maximum value that \( r \) can achieve (and the radius of the largest loops).
• Period: With a factor of \( \frac{1}{3} \) within the sine function, the period extends such that the function completes one full cycle as \( \theta \) goes from 0 to \( 6\pi \). This means that the pattern of the curve repeats thrice as \( \theta \) moves from 0 to \( 6\pi \).

The sine function causes the described curve to form repeating petal shapes, due to its cyclic nature.

Sketching Polar Graphs

Sketching polar graphs involves plotting the function by determining key points and recognizing patterns. This process can reveal intricacies within the graph, making it an opportunity to explore mathematical beauty.

For the polar equation \( r = 4 \sin\left(\frac{\theta}{3}\right) \), here’s a step-by-step process to sketch the curve:

• Identify Key Points: Calculate \( r \) for specific values of \( \theta \) such as 0, \( \pi \), and multiples up to \( 6\pi \). Notably, at these points, the sine function equals zero, making \( r = 0 \), indicating that the curve touches the pole.
• Determine Amplitude and Period: Understanding the maximum radial distance (4 units in this case) and the curve's repetition helps in sketching accurate loops.
• Draw Smooth Curves: Use these points to plot the curve, observing the wave created by the sine function between each evaluated point, which forms loops or petals around the pole.

Each visual element depicted can be traced back to the properties of the sine function, and by embracing this connection, sketching becomes a delightful uncovering of symmetry and periodicity.