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Chapter 6: Problem 57

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

Short Answer

Expert verified
The graph of \(r = 5 - 4\sin \theta\) is a dimpled lima莽on with no symmetry.

Step by step solution

01

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.
02

Recognize the Form and General Shape

The equation \(r = 5 - 4 \sin \theta\) is a type of lima莽on. When a polar equation has the form \(r = a \pm b \sin \theta\) or \(r = a \pm b \cos \theta\), it represents a lima莽on. Here, \(a = 5\) and \(b = 4\), which implies that the curve is a dimpled lima莽on because \(a > b\) but \(a < 2b\).
03

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鈥檛 simplify back to the original.So, this graph doesn't seem to have any clear symmetry.
04

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 lima莽on starting at \(r = 5\), peaking at \(r = 9\) and repeating similarly as \(\theta\) increases.
05

Connect the Points Smoothly

Connect the points obtained in Step 4 smoothly to illustrate a dimpled lima莽on. 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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

尝颈尘补莽辞苍
In the realm of polar coordinates, a lima莽on 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 lima莽on because it follows the form \( r = a \pm b \sin \theta \) where \( a > b \) but \( a < 2b \).
  • When \( a < b \), the lima莽on 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 lima莽on.
These conditions help us predict its appearance even before sketching. For our particular problem, the lima莽on 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 lima莽on lacks standard symmetries commonly seen in simpler polar graphs such as circles or roses.
Graph Sketching
After understanding the nature of the lima莽on 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 lima莽on. 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.

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Most popular questions from this chapter

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.\(r=4 \cos \theta ;\left(2, \frac{\pi}{3}\right)\)

Find the points at which the following polar curves have a horizontal or vertical tangent line.\(r=2 \sin (2 \theta)\)

Without using technology, sketch the polar curve \(\theta=\frac{2 \pi}{3}\).

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), so the polar equation \(r=f(\theta)\) is now written in parametric form.\(r=4 \cos (2 \theta) ;\) tips of the leaves

For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. Find \(\frac{d y}{d x}, \frac{d x}{d y}\), and \(\frac{d^{2} x}{d y^{2}}\) of \(y=\left(2+e^{-t}\right), x=1-\sin (t)\)
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