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Chapter 6: Problem 72

Use a graphing utility to graph the polar equation. $$r=3 \sin \left(\theta+\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The polar equation results in a sinusoidal curve with an amplitude of 3 units, shifted to the left by \(\frac{\pi}{4}\) units and repeating every 2蟺 units. This can be graphed using a dedicated graphing utility.

Step by step solution

01

Understanding the Polar Equation

The first step is to understand what the equation stands for. It is a sinusoidal function, with a phase shift of \(\frac{\pi}{4}\) to the left and an amplitude of 3.
02

Plotting Using Graphing Utility

In a graphing utility, switch the graph from Cartesian mode to Polar mode. Then, plug in the polar equation \(r=3 \sin \left(\theta+\frac{\pi}{4}\right)\). The utility will automatically graph the polar equation.
03

Interpreting the Resulting Graph

Observe the graph on the screen. You should see a sinusoidal curve shifted to the left by \(\frac{\pi}{4}\). The maximum value the curve reaches is 3, as indicated by the number in front of the sine function. The curve restarts itself every 2蟺, reflecting the periodicity of the sine function.

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Most popular questions from this chapter

Use a graphing utility to graph each butterfly curve. Experiment with the range setting, particularly \(\theta\) step, to produce a butterfly of the best possible quality. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}, \quad \mathbf{w}=-6 \mathbf{i}-9 \mathbf{j}$$

Find \(\text {pro}_{\mathbf{w}} \mathbf{V}\) Then decompose v into two vectors, \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}.\) $$\mathbf{v}=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=3 \mathbf{i}+6 \mathbf{j}$$

What are orthogonal vectors?

Find a value of \(b\) so that \(15 \mathbf{i}-3 \mathbf{j}\) and \(-4 \mathbf{i}+b \mathbf{j}\) are orthogonal.
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