Problem 39 Sketch the graphs of the polar e... [FREE SOLUTION]

Chapter 10: Problem 39

Sketch the graphs of the polar equations. Indicate any symmetries around either coordinate axis or the origin. \(r=2 \cos \theta \quad\) (circle)

Short Answer

Expert verified

The graph is a circle centered at (1, 0) with a radius of 1, symmetric about the x-axis.

Step by step solution

Recognize the Polar Equation Form

The given polar equation is \( r = 2 \cos \theta \). This is a standard form of a polar equation that represents a circle.

Convert Polar to Cartesian Coordinates (Optional)

To better understand the graph, you might convert the polar equation to Cartesian coordinates. Using the polar to Cartesian conversion formulas, \( x = r \cos \theta \) and \( y = r \sin \theta \), substitute \( r = 2 \cos \theta \):\( x = (2 \cos \theta) \cos \theta = 2 \cos^2 \theta \) and \( y = 2 \cos \theta \sin \theta \). From this, we know that \( r \cos \theta = x \), so \( x = 2 \cos^2 \theta \) gives the equation \( x = x^2 + y^2 \), confirming a circle.

Identify Symmetry

Observe symmetry properties in polar coordinates. For \( r = 2 \cos \theta \), if the equation remains unchanged when \( \theta \) is replaced with \( -\theta \), the graph is symmetric about the x-axis. Here, replacing \( \theta \) with \( -\theta \) results in the same equation, confirming x-axis symmetry. There is no y-axis or origin symmetry due to the cosine function.

Determine Key Points

Find key points by substituting specific angles. For example, at \( \theta = 0 \), \( r = 2 \), which is a point on the positive x-axis. At \( \theta = \frac{\pi}{2} \), \( r = 0 \), which corresponds to the pole. At \( \theta = \pi \), \( r = -2 \), interpreted as a point at (2, 0) on the circle.

Sketch the Graph

Use the determined key points and symmetry information to sketch the graph. The equation \( r = 2 \cos \theta \) describes a circle of radius 1 with its center at (1, 0) in Cartesian coordinates. Draw a circle centered at (1, 0) with a radius of 1.

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Key Concepts

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

Cartesian Coordinates

To fully understand the graph of the equation \( r = 2 \cos \theta \), we can convert it from polar to Cartesian coordinates. This gives us a picture we are more familiar with.
The polar to Cartesian conversion has some handy formulas:

\( x = r \cos \theta \)
\( y = r \sin \theta \)

For the equation \( r = 2 \cos \theta \), substituting into these formulas means:

\( x = (2 \cos \theta) \cos \theta = 2 \cos^2 \theta \)
\( y = 2 \cos \theta \sin \theta \)

Now knowing that \( r \cos \theta = x \), substitute the expression for \( x \). This gives \( x = x^2 + y^2 \), a familiar equation representing a circle centered at \((1,0)\).
The conversion from polar to Cartesian coordinates helps us see how the circle in polar form maps to one we recognize on a Cartesian plane.

Symmetry in Polar Graphs

Symmetry in polar graphs simplifies their understanding and sketching. For the given polar equation \( r = 2 \cos \theta \), let's explore these symmetries step by step.
Polar graphs can be symmetric about:

The x-axis
The y-axis
The origin

For symmetry about the x-axis, replace \( \theta \) with \( -\theta \) in our equation. If it remains unchanged, x-axis symmetry is confirmed.
Let's test it:

\( r = 2 \cos(\theta) \)
\( r = 2 \cos(-\theta) \)

Since these are equal, the graph shows x-axis symmetry. However, the equation does not remain unchanged with substitutions for y-axis or origin symmetries, thus they are not present in this case.
Recognizing these patterns helps clarify the graph's properties, aiding accurate plotting and comprehension.

Graph Sketching Techniques

Graph sketching is like painting a picture, where understanding key details can make it easier. Let's take a closer look at sketching the given equation \( r = 2 \cos \theta \).
Begin by determining critical points. These are specific angles where the output \( r \) gives key positions on the graph:

At \( \theta = 0 \), \( r = 2 \); this point is on the positive x-axis.
At \( \theta = \frac{\pi}{2} \), \( r = 0 \); this point is the pole, or center of the graph.
At \( \theta = \pi \), \( r = -2 \); in polar coordinates, this reflects back to point \((2,0)\) on the positive x-axis.

Using these points helps anchor the graph.
The equation describes a circle of radius 1, centered at (1,0) in Cartesian terms.
Use the x-axis symmetry discovered earlier to mirror points accurately across the axis. Combine this information to confidently sketch a neat circle centered at (1,0) with a radius of 1, ensuring the graph correctly represents the equation.

91影视

Sketch the graphs of the polar equations. Indicate any symmetries around either coordinate axis or the origin. \(r=2 \cos \theta \quad\) (circle)

Short Answer

Step by step solution

Recognize the Polar Equation Form

Convert Polar to Cartesian Coordinates (Optional)

Identify Symmetry

Determine Key Points

Sketch the Graph

Key Concepts

Cartesian Coordinates

Symmetry in Polar Graphs

Graph Sketching Techniques

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