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Chapter 11: Problem 4

Identify the symmetries of the curves.Then sketch the curves in the \(x y\) plane. $$r=1+\sin \theta$$

Short Answer

Expert verified
The curve is symmetric about the vertical axis \(\theta = \frac{\pi}{2}\) and is a cardioid.

Step by step solution

01

Identify the Equation Form

The given polar equation is given by \[r = 1 + \sin \theta\]This is a well-known form known as a cardioid. A cardioid is symmetric with respect to the vertical axis (line \(\theta = \frac{\pi}{2}\)).
02

Determine Symmetries

Check for symmetry with respect to the polar axis (horizontal axis). If replacing \(\theta\) with \(-\theta\) gives the original equation, it has polar axis symmetry. \[1 + \sin(-\theta) = 1 - \sin(\theta)\]This is not equal to the original equation, hence the curve is not symmetric about the polar axis.Next, check symmetry with respect to the line \(\theta = \frac{\pi}{2}\). Replace \(\theta\) with \(\pi - \theta\):\[1 + \sin(\pi - \theta) = 1 + \sin \theta\]This confirms vertical line symmetry.
03

Sketch the Curve

To sketch the cardioid, recognize that \(r = 1 + \sin \theta\) implies that for \(\theta = 0\), \(r = 1\); for \(\theta = \frac{\pi}{2}\), \(r = 2\); for \(\theta = \pi\), \(r = 1\); and for \(\theta = \frac{3\pi}{2}\), \(r = 0\). This will appear as a heart-shaped curve centered around the origin, peaking at the top.

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Key Concepts

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

Curve Symmetry
Symmetry in polar coordinates can significantly simplify the process of graphing a curve. It helps us understand the overall shape without needing to plot every point individually. There are several types of symmetries
  • Polar Axis Symmetry - Symmetry about the horizontal axis. For a curve to have this symmetry, replacing \( \theta \) with \( -\theta \) should yield the same equation. In this exercise, the equation becomes \( 1 - \sin(\theta) \), which is different from the original. Thus, there is no polar axis symmetry.
  • Line \( \theta = \frac{\pi}{2} \) Symmetry - This symmetry refers to a vertical line symmetry. Substitute \( \theta \) with \( \pi - \theta \). We see that \( 1 + \sin(\pi - \theta) = 1 + \sin \theta \), so the symmetry matches, confirming that the curve has this vertical line symmetry.
  • Symmetry about the Pole - This means rotational symmetry around the center. To check this, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \). For this exercise, the result does not match, indicating no pole symmetry.
Recognizing symmetry helps to understand the structure and make sketching easier.
Cardioid
A cardioid is a special type of polar curve that resembles a heart shape. It is one of the simplest polar equations and takes the form \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). In the given exercise, we have \( r = 1 + \sin \theta \), which aligns with the standard cardioid form with both \( a \) and \( b \) equal to 1. This results in specific properties:
  • Shape - As previously mentioned, the graph resembles a heart shape centered at the pole (origin).
  • Loop - The cardioid has a single loop extending from the pole outward.
  • Maximum and Minimum - With this form, the maximum value is \( r = 2 \) when \( \theta = \frac{\pi}{2} \), and the minimum is \( r = 0 \) when \( \theta = \frac{3\pi}{2} \).
This simple structure of the cardioid makes it an excellent introductory example for understanding polar curves.
Sketching Polar Curves
Sketching a polar curve involves understanding how \( r \) changes as \( \theta \) varies. To begin with, it鈥檚 useful to identify key angles that help form the shape of the curve. Here is a step-by-step method to sketch the cardioid:
  • Select Key Angles: Choose angles like \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
  • Calculate \( r \) for each angle: Using the exercise example, plug in these angles one by one to find \( r \):
    • \( r(0) = 1 + \sin(0) = 1 \)
    • \( r\left( \frac{\pi}{2} \right) = 1 + \sin\left( \frac{\pi}{2} \right) = 2 \)
    • \( r(\pi) = 1 + \sin(\pi) = 1 \)
    • \( r\left( \frac{3\pi}{2} \right) = 1 + \sin\left( \frac{3\pi}{2} \right) = 0 \)
  • Plot Points and Connect: On polar graph paper, plot these points and smoothly connect them to observe the cardioid shape.
Using these techniques helps you visualize and draw complex shapes with confidence, easing the learning of polar coordinate systems.

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Most popular questions from this chapter

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=1, \quad y=2$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=2 \cos \theta+2 \sin \theta$$

Give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$e=2, \quad x=4$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \left(\frac{2 \pi}{3}-\theta\right)=5$$

Replace the Cartesian equations with equivalent polar equations. $$x^{2}+(y-2)^{2}=4$$
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