Question

Sketch a graph of the polar equation and identify any symmetry. $$ r=5-4 \sin \theta $$

Short answer

The graph of \(r = 5 - 4\sin \theta\) is a dimpled limaçon with no symmetry.

Step-by-step solution

Understand the Polar Equation

The given polar equation is \(r = 5 - 4 \sin \theta\). This equation describes a curve in the polar coordinate system, where \(r\) is the radius and \(\theta\) is the angle in radians.

Recognize the Form and General Shape

The equation \(r = 5 - 4 \sin \theta\) is a type of limaçon. When a polar equation has the form \(r = a \pm b \sin \theta\) or \(r = a \pm b \cos \theta\), it represents a limaçon. Here, \(a = 5\) and \(b = 4\), which implies that the curve is a dimpled limaçon because \(a > b\) but \(a < 2b\).

Test for Symmetry

To test for symmetry, we will check against the polar axes. - Symmetry with respect to the x-axis: Replace \(\theta\) by \(-\theta\) and see if the equation remains unchanged. \(r = 5 - 4 \sin(-\theta) = 5 + 4 \sin \theta\) is not the same.- Symmetry with respect to the y-axis: Replace \(r\) by \(-r\) and \(\theta\) by \(\pi - \theta\) to see if the equation is unchanged. The transformed equation doesn't simplify back to the original.- Symmetry with respect to the origin: Replace \(\theta\) by \(\theta + \pi\) and \(r\) by \(-r\). The equation doesn’t simplify back to the original.So, this graph doesn't seem to have any clear symmetry.

Sketch the Graph Using Key Angles

To sketch the graph, calculate \(r\) for key angles \(\theta\):- At \(\theta = 0\), \(r = 5\) and when \(\theta = \frac{\pi}{2}\), \(r = 1\).- At \(\theta = \pi\), \(r = 5 + 0 = 5\), and when \(\theta = \frac{3\pi}{2}\), \(r = 9\).These give a series of points: as \(\theta\) increases from 0 to 2\(\pi\), the values of \(r\) change, forming a dimpled limaçon starting at \(r = 5\), peaking at \(r = 9\) and repeating similarly as \(\theta\) increases.

Connect the Points Smoothly

Connect the points obtained in Step 4 smoothly to illustrate a dimpled limaçon. The graph has a flat indentation at \(\theta = \frac{\pi}{2}\), loops outward again, and follows a similar path from \(\theta = \pi\) to \(2\pi\). Use a plotting tool or graph paper to ensure accuracy in connecting the points.

Key concepts

Limaçon

In the realm of polar coordinates, a limaçon is a fascinating type of curve. Derived from the French word for "snail," it presents forms that may either have a loop, a cusp, or simply a dimple, depending on the parameters in its equation. The equation in our problem is given by \( r = 5 - 4 \sin \theta \). This can be classified as a dimpled limaçon because it follows the form \( r = a \pm b \sin \theta \) where \( a > b \) but \( a < 2b \).

• When \( a < b \), the limaçon has an inner loop.
• If \( a = b \), it presents a cardioid shape, similar to a heart.
• For \( a > b \) but \( a < 2b \), it becomes a dimpled limaçon.

These conditions help us predict its appearance even before sketching. For our particular problem, the limaçon does not have an inner loop and instead features a characteristic dimple.

Symmetry Analysis

Understanding symmetry in polar graphs can dramatically simplify your analysis and graphing process. For our equation \( r = 5 - 4 \sin \theta \), the symmetry tests involve transformations that may reveal whether the curve mirrors across any axes or points:

• X-axis Symmetry: Replace \(\theta\) with \(-\theta\) and check if the equation remains unchanged. The transformation yields \( r = 5 + 4 \sin \theta \), which differs from the original.
• Y-axis Symmetry: Check this by replacing \( r \) with \( -r \) and \( \theta \) with \( \pi - \theta \). No simplification leads back to the initial equation.
• Origin Symmetry: Use \( \theta + \pi \) for \( \theta \) and \( -r \) for \( r \). Here again, the equation does not revert to its starting form.

Given these analyses, the limaçon lacks standard symmetries commonly seen in simpler polar graphs such as circles or roses.

Graph Sketching

After understanding the nature of the limaçon and checking for symmetry, you can proceed to sketch the polar graph. Begin by calculating the value of \( r \) at key angles:

• At \( \theta = 0 \), \( r = 5 \).
• At \( \theta = \frac{\pi}{2} \), \( r = 1 \), which is the minimum.
• At \( \theta = \pi \), \( r = 5 \).
• At \( \theta = \frac{3\pi}{2} \), \( r = 9 \), reaching its maximum.

Plotting these key points provides a framework for the curve. As \( \theta \) progresses from \( 0 \) to \( 2\pi \), gently connect these points to form the dimpled limaçon. You will notice an indentation around \( \theta = \frac{\pi}{2} \). Using graph paper or plotting software can help achieve a smoother and more accurate curve to showcase its unique dimpled structure.