Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 14
$$ \text { Graph each equation. } $$ $$ r=3 \cos \theta $$
Short Answer
Expert verified
The graph of \( r = 3 \cos \theta \) is a circle centered at \( (1.5, 0) \) with a radius of 1.5 units.
Step by step solution
01
Understand the Polar Equation
The given equation is in polar form: \( r = 3 \cos \theta \). In polar coordinates, \( r \) is the distance from the origin, and \( \theta \) is the angle from the positive x-axis. Here, \( r \) is expressed as a function of \( \theta \).
02
Identify the Type of Polar Graph
The equation \( r = 3 \cos \theta \) represents a circle in polar coordinates. Generally, equations of the form \( r = a \cos \theta \) or \( r = a \sin \theta \) describe circles. In this equation, \( a = 3 \), indicating a circle.
03
Determine Properties of the Circle
For \( r = a \cos \theta \), the circle is centered at \( (\frac{a}{2}, 0) \) in Cartesian coordinates, with a radius of \( \frac{a}{2} \). Here, \( a = 3 \), so the center is \( (1.5, 0) \) and the radius is \( 1.5 \).
04
Plot the Graph
To plot the graph, draw a circle centered at \( (1.5, 0) \) with a radius of 1.5 units. In polar terms, plot several points at various angles \( \theta \) and use the equation \( r = 3 \cos \theta \) to find the corresponding \( r \) value, ensuring these points lie on the circle.
05
Verify and Label the Graph
Check a few more points, such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), to ensure they satisfy the equation and fall on the circle plotted. Label the graph with the equation \( r = 3 \cos \theta \) and mark the center and radius with their respective coordinates and measurements.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Equations
Polar equations are a way of expressing curves using polar coordinates rather than the typical Cartesian coordinates. In polar coordinates, every point is determined by a distance from a reference point, called the pole (which is similar to the origin in Cartesian coordinates), and an angle from a reference direction, usually the positive x-axis.
- In polar form, we have the variables \(r\) and \(\theta\), where \(r\) is the radius or distance from the pole.
- \(\theta\) is the angle in radians measured from the positive x-axis, moving counterclockwise.
- The equation \(r = 3 \cos \theta\) is an example of a polar equation, where \(r\) is expressed in terms of \(\theta\).
Graphing Circles
Graphing a circle using polar coordinates can be a different experience than using Cartesian coordinates. For the equation \(r = 3 \cos \theta\), which represents a circle, a few key insights can be drawn.
- First, recognize the form \(r = a \cos \theta\) or \(r = a \sin \theta\) generally describes a circle in polar coordinates.
- The value of \(a\) determines both the radius and the center's position of the circle. In our equation, \(a = 3\).
- The circle is centered at \((\frac{a}{2}, 0)\) in Cartesian coordinates with a radius of \(\frac{a}{2}\). Thus, the center is at \((1.5, 0)\) with a radius of 1.5 units.
Conversion to Cartesian Coordinates
Converting polar coordinates to Cartesian coordinates sometimes makes graphing or algebraic manipulation more straightforward. Let's see how this conversion works.
- The conversion formulas between polar and Cartesian coordinates are: \(x = r \cos \theta\) and \(y = r \sin \theta\).
- For the polar equation \(r = 3 \cos \theta\), substitute \(r\) and \(\theta\) into the conversion formulas.
- Since \(r = 3 \cos \theta\), then \(x = r \cos \theta = (3 \cos \theta) \cos \theta = 3 \cos^2 \theta\), giving \(x = r = 3 \cos \theta\).
- We end with a new equation such as \( (x - 1.5)^2 + y^2 = 1.5^2 \), defining the same circle in Cartesian coordinates.
