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Chapter 6: Problem 50

Sketch the graph of each polar equation. $$ r=1-\cos \theta(\text { cardioid }) $$

Short Answer

Expert verified
Sketch the cardioid by plotting points for \( r = 1 - \cos \theta \) at key angles and connecting them smoothly.

Step by step solution

01

- Understand the Polar Equation

The given polar equation is \( r = 1 - \cos \theta \). Identify the type of graph it represents, which is a cardioid.
02

- Calculate Key Points

Calculate the values of \( r \) for key angles \( \theta \). For example, when \( \theta = 0 \): \[ r = 1 - \cos(0) = 1 - 1 = 0 \] when \( \theta = \frac{\pi}{2} \): \[ r = 1 - \cos \left( \frac{\pi}{2} \right) = 1 - 0 = 1 \] and so on.
03

- Plot the Points

Plot the calculated points in the polar coordinate system. Start from \( \theta = 0 \) and continue to \( \theta = 2\pi \). This will sketch the cardioid shape.
04

- Connect Points Smoothly

Connect the plotted points smoothly to reveal the full cardioid curve. Make sure to check points like where \( \theta \) is \( \pi \) and \( 2\pi \) to ensure accuracy.
05

- Verify the Shape

Double-check that your graph resembles a typical cardioid shape, which should have a single cusp at \( \theta = 0 \) and be symmetric with respect to the line \( \theta = \pi \).

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Key Concepts

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

Polar Coordinates
Polar coordinates are an alternative to the Cartesian coordinate system. They represent points in a plane using a distance and an angle. Instead of using coordinates like \( (x, y) \), polar coordinates use \( (r, \theta) \).
Here:
  • \(r\) is the distance from the origin (also called the radial coordinate).
  • \(\theta\) is the angle from the positive x-axis (also called the angular coordinate).
This system is especially useful for equations involving circles and other curves that are easier to describe with angles and distances from a central point.
For instance, the point that is 3 units away from the origin at an angle of 45掳 can be written as \( (r = 3, \theta = 45掳) \).
It makes some equations and graphs simpler, much like in our exercise with the cardioid.
Cardioid
A cardioid is a heart-shaped plot you get from specific polar equations. The name 'cardioid' stems from the Greek word for 'heart.'
Our equation, \( r = 1 - \cos \theta \), generates a cardioid.
This type of polar graph typically has a prominent cusp at one point. Here are some features of a cardioid:
  • The cardioid is symmetric with respect to the line where \theta = \pi\.
  • It touches the origin when \( \theta = 0 \).
    In our example, when \theta = 0\, \( r\) = 0 as well.
  • The farthest point from the origin happens when \theta = \pi\.
Cardioids are essential in different fields like physics and engineering. Their property of reflecting angles more than 90 degrees makes them useful in designing acoustically sensitive spaces.
Plotting Points
Plotting points in polar coordinates requires calculating the \( r \) value for various angles \( \theta \). Here鈥檚 a simple way to do it:
  • Choose some key angles, such as \( \theta = 0, \pi/6, \pi/4, \pi/2, \pi, ... etc. \).
  • Plug these angles into your original polar equation (like \( r = 1 - \cos \theta \)) to find the corresponding \( r \) values.
  • Mark these \( r, \theta \) pairs on your polar grid.
For our cardioid equation, when \theta = 0\, \( r = 1 - \cos(0) = 0 \).
When \ \theta = \pi/2\, \( r=1- \cos (\frac{\,\theta}{2}) = 1 - 0 = 1 \). This shows you how plotting works.
After you plot each key point, connect them smoothly for a clear shape. Double-checking helps ensure your cardioid looks right.

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Most popular questions from this chapter

For each polar equation, write an equivalent rectangular equation. $$ \theta=\frac{\pi}{4} $$

Graph each pair of polar equations on the same screen of your calculator and use the trace feature to estimate the polar coordinates of all points of intersection of the curves. Check your calculator manual to see how to graph polar equations. $$ r=\sin \theta, r=\sin 2 \theta $$

Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ (-2+4 i) \div(-i) $$

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Write each expression in the form \(a+\) bi where \(a\) and \(b\) are real numbers. $$ i^{34}+i^{9} $$
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