Problem 28 Test for symmetry and then graph... [FREE SOLUTION]

Chapter 6: Problem 28

Test for symmetry and then graph each polar equation. $$r=4 \cos 3 \theta$$

Short Answer

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The given polar equation, $r=4 \cos 3 \theta$, is symmetric about the x-axis, y-axis, and origin. The graph consists of 3 petals formed due to the factor 3 in the function, pointing in directions based on the calculated table of values going from $\theta = 0$ to $\theta = 2\pi$.

Step by step solution

Test for Symmetry

The given equation is $r=4 \cos 3 \theta$. We test for symmetry by replacing $\theta$ by $-\theta$ for symmetry about the x-axis, replacing $\theta$ by $\pi - \theta$ for symmetry about the y-axis, and replacing $r$ by $-r$ for symmetry about the origin. On performing these, the original equation remains unchanged thus it has symmetry about the x-axis, y-axis and the origin.

Create a Table of Values

Choose selective values for $\theta$. Sub these values into $r = 4 \cos 3\theta$ to get the corresponding $r$. For example, if we choose $\theta = 0$, then $r = 4 \cdot \cos(0) = 4$. Repeat this method for different values of $\theta$. Create a table for these chosen values of $\theta$ and corresponding computed $r$.

Sketch the Polar Curve

Plot the points from the table on the polar graph, make sure the $r$-values are plotted in the correct direction associated with their respective $\theta$-values. Connect the points to complete the graph. Depending on the symmetries determined in step 1, reflect the plotted points accordingly over the x and y axes.

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Key Concepts

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

Symmetry in Graphs

Symmetry in graphs is an important concept when dealing with polar coordinates and equations. It helps us understand how the graph of an equation behaves and can simplify graphing by reducing the number of points we need to plot. There are three types of symmetries that are typically tested:

Symmetry about the x-axis: This involves replacing $ \theta $ with $ -\theta $. If the equation remains unchanged, the graph is symmetrical around the x-axis.
Symmetry about the y-axis: This test is done by replacing $ \theta $ with $ \pi - \theta $. If the equation stays the same, it means the graph is symmetric around the y-axis.
Symmetry about the origin: By replacing $ r $ with $ -r $, check if the equation is unchanged. If it is, then the graph is symmetric about the origin.

Testing our polar equation $ r=4 \cos 3\theta $, it shows symmetry across all three divisions, which can greatly simplify graphing and understanding the shape of its curve.

Trigonometric Functions

Trigonometric functions play a crucial role in polar equations. They relate the angle $ \theta $ to distances in the plane. In our specific equation, $ r=4 \cos 3\theta $, we see the cosine function being used. Understanding cosine helps us determine the behavior of the graph:

Cosine Function Basics: The cosine function ranges between -1 and 1. In polar graphs, the value of $ r $ changes with angles as per the cosine function, affecting the distance a point is from the origin.
Frequency and Period: The expression $ 3\theta $ indicates the frequency of oscillation of the cosine function, meaning the graph will repeat its pattern every $ \frac{2\pi}{3} $.
Amplitude: Here, the amplitude is 4, indicating the maximum distance from the origin that $ r $ can attain.

Understanding these aspects of trigonometric functions allows us to predict and correctly sketch the behavior of polar graphs.

Graphing Polar Equations

Graphing polar equations involves translating a set of radial points into a visual representation in the polar coordinate plane. The process typically starts by identifying symmetries, followed by calculating select points, and plotting them:

Select Values for $ \theta $: Start by choosing angles that evenly spread across the polar plane, like $ 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \pi, $ and so on.
Calculate Corresponding $ r $-values: Substitute these angle values into the equation $ r=4 \cos 3\theta $ to find the radial distance of each point from the origin.
Plot Points on Polar Graph: Using the polar coordinates system, plot each point by placing them at the calculated distance from the origin at their respective angles.
Connect Points & Apply Symmetry: Draw smooth curves connecting all the plotted points, and utilize identified symmetries to reflect parts of the graph as needed.

By considering symmetries and understanding the trigonometric influences, the graph can be sketched accurately and efficiently, providing visual insights into the behavior of the polar equation.

91影视

Test for symmetry and then graph each polar equation. $$r=4 \cos 3 \theta$$

Short Answer

Step by step solution

Test for Symmetry

Create a Table of Values

Sketch the Polar Curve

Key Concepts

Symmetry in Graphs

Trigonometric Functions

Graphing Polar Equations

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