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Chapter 6: Problem 32

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

Short Answer

Expert verified
The polar equation \(r=3+\sin \theta\) is symmetric about the y-axis, but not about the x-axis or the origin. The graph of the equation forms a cardioid shape.

Step by step solution

01

Symmetry Tests

Apply the tests to see if the given polar equation \(r=3+\sin \theta\) is symmetric about the x-axis, the y-axis, or the origin. 1. Test for symmetry about the x-axis (replace \(\theta\) by \(-\theta\)). If the equation remains unchanged, then it is symmetric about the x-axis. 2. Test for symmetry about the y-axis (replace \(\theta\) by \(\pi-\theta\)). If the equation remains unchanged, then it is symmetric about the y-axis. 3. Test for symmetry about the origin (replace \(r\) by \(-r\)). If the equation remains unchanged, then it is symmetric about the origin.
02

Apply the Tests

Replace \(\theta\) with \(-\theta\), \(\pi-\theta\) and \(r\) with \(-r\) in the equation \(r=3+\sin \theta\), and observe the results: 1. With \(\theta=-\theta\), the equation becomes \(r=3+\sin(-\theta)\) which simplifies to \(r=3-\sin \theta\), so it is not symmetric about the x-axis. 2. With \(\theta=\pi-\theta\), the equation becomes \(r=3+\sin(\pi-\theta)\) which simplifies to \(r=3+\sin \theta\), the equation remains unchanged, so it is symmetric about the y-axis. 3. With \(r=-r\), we can't replace \(r\) with \(-r\) in our equation since it results in \(-r=3+\sin \theta\) which is not equivalent to our original equation. Therefore, it is not symmetric about the origin.
03

Graph the Polar Equation

The equation is symmetric about the y-axis. To graph this, we need to plot appropriate values of \(r\) for each \(\theta\). For example, the values of \(r\) for key angles: \(0, \pi/2, \pi, 3\pi/2, 2\pi\) are: \(3+sin0=3, 3+sin\pi/2 = 4, 3+sin\pi = 3, 3+sin3\pi/2=2, 3+sin2\pi=3\) respectively. Mark these values in a polar grid. The plotting of given key angles will form a cardioid shape, showing the graph of the equation.

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Most popular questions from this chapter

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=2 \sin 3 \theta, r_{2}=2 \sin 3\left(\theta+\frac{\pi}{6}\right)$$

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+10 \mathbf{j}$$

The components of \(\mathbf{v}=180 \mathbf{i}+450 \mathbf{j}\) represent the respective number of one-day and three-day videos rented from a video store. The components of \(\mathbf{w}=3 \mathbf{i}+2 \mathbf{j}\) represent the prices to rent the one-day and three-day videos, respectively. Find \(\mathbf{v} \cdot \mathbf{w}\) and describe what the answer means in practical terms.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=\mathbf{i}+\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$$
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