Problem 38 Test for symmetry and then graph... [FREE SOLUTION]

Chapter 6: Problem 38

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

Short Answer

Expert verified

The given polar equation $r=4 \cos \theta+4 \sin \theta$ is symmetric with respect to the origin but not with respect to $\theta= \frac{\pi}{2}$ or the pole. The graph of this equation will be a circle centred at the pole.

Step by step solution

Symmetry tests

First, test the given polar equation for symmetry. For symmetry with respect to the line $\theta = \frac{\pi}{2}$, replace $\theta$ with $-\theta$ and check if you can simplify it back to the original equation. For symmetry about the pole, replace $r$ with $-r$, and change $\theta$ to $\theta + \pi$. For symmetry about the line $\theta=0$ or $\theta=\pi$, replace $\theta$ with $\pi - \theta$.

Graph the polar equation

After testing for symmetry, plot the polar equation on a graph. Find the representative values of $r$ by substituting some possible values of $\theta$. Start by inserting multiples of $\frac{\pi}{4}$ and sketch the corresponding coordinates. Further, connect these points smoothly to draw the entire graph.

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Key Concepts

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

Symmetry in Polar Equations

Understanding symmetry in polar equations can simplify the graphing process and tell us a lot about the equation's behavior.
Symmetry testing involves checking if the graph looks the same when it's reflected over certain lines or about points.

Symmetry with respect to the polar axis: Check this by replacing $ \theta $ with $-\theta $. If the equation remains unchanged, the graph is symmetric around the polar axis (line $ \theta = 0 $).
Symmetry with respect to the line $ \theta = \frac{\pi}{2} $: This symmetry is verified by testing if $ r(\pi - \theta) $ results in an unchanged equation. This means the graph looks the same when flipped vertically over this line.
Symmetry about the pole: Replace $ r $ with $-r $ and $ \theta $ with $ \theta+\pi $. If the equation doesn't change, it's symmetric around the origin.

Applying these tests helps determine how our graph will appear, reducing repetition and giving a structured approach for sketching it.

Graphing Polar Equations

Graphing polar equations is a bit different compared to Cartesian graphs, but it's straightforward once you get the hang of it. Polar equations are plotted in a polar coordinate system, where each point is determined by the distance from the origin ($ r $) and the angle ($ \theta $).Here's a simple way to graph a polar equation:

Choose values for $ \theta $: Start with key angles like $ 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}, \pi $, etc.
Calculate $ r $ for each $ \theta $: Substitute these angles into your polar equation to find the corresponding $ r $ values.
Plot the points: On polar graph paper, each point is located by moving $ r $ units from the origin at an angle of $ \theta $.
Connect the points: Use a smooth curve to connect these plotted points. This will give you a visual representation of the polar equation.

This method provides a systematic way to derive the graph of any polar equation, giving you insight into its geometric representation.

Polar Equation $r=4 \cos \theta+4 \sin \theta$

The given polar equation $ r = 4 \cos \theta + 4 \sin \theta $ can be explored through both symmetry and graphing techniques. This particular equation blends both cosine and sine components, which makes it interesting to analyze.**Handling Symmetry:**

To check its symmetry, we employ the previously discussed methods.Replace $ \theta $ with $-\theta$ or $ \pi - \theta $ and test for the same form.Similarly, altering $ r $ to $-r $ can provide clues on whether it has pole symmetry.
Throughout these tests, verify whether the simplified equation corresponds to the original.

**Graphing the Equation:**

Starting at specific angles, for instance, multiples of $ \frac{\pi}{4} $ provide a handy set of points.Calculate $ r $ values for these angles.
Plot these points on polar graph paper, maintaining proper attention to accurate distance and angle representation.
When connecting the points, look for a circular or elliptical shape which is a key feature of equations mixing $ \cos \theta $ and $ \sin \theta $.

By delving into these techniques, students can securely understand the structure and appearance of this polar equation, aiding both analytical and graphical perspectives.

91影视

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

Short Answer

Step by step solution

Symmetry tests

Graph the polar equation

Key Concepts

Symmetry in Polar Equations

Graphing Polar Equations

Polar Equation \(r=4 \cos \theta+4 \sin \theta\)

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