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Chapter 6: Problem 73

Use a graphing utility to graph the polar equation. $$r=2 \cos \left(\theta-\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The graph of the polar equation \(r=2 \cos (\theta-\frac{\pi}{4})\) is a circle with a radius of 2, which is shifted \(\frac{\pi}{4}\) radians in the counter-clockwise direction.

Step by step solution

01

Recognizing the form of polar equations

As the polar equation given is \(r=2 \cos \left(\theta-\frac{\pi}{4}\right)\), it is clear that the equation presents a circle shifted and scaled. Here, 'r' represents the distance from the origin, '2' denotes how large the circle is (radius), and \(\frac{\pi}{4}\) is the shift in direction clockwise of the circle.
02

Choose a graphing utility

Now the task is to graph the given polar equation. Suitable utility programs for this include Desmos, Grapher on Mac, or TI calculators. It doesn't matter what technology you use as long as you know how to input it correctly.
03

Inputting the equation

Enter the equation into the graphing utility and adjust the viewing window appropriately, to capture the complete graph. Larger polar angles range from 0 to \(2\pi\) radians or 0 to 360 degrees. Change the settings to polar mode or use polar function to input the given equation: \(r=2 \cos (\theta - \frac{\pi}{4})\), as per the graphing utility instructions.
04

Graphing and interpreting the result

As you graph the equation, you'll see a circle with a radius of 2 shifted by \(\frac{\pi}{4}\) radians in the counter-clockwise direction from the initial line polar axis (usually, right horizontal direction is considered as the initial line).

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Most popular questions from this chapter

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 \(\mathbf{w}.\) $$\mathbf{v}=2 \mathbf{i}+4 \mathbf{j}, \quad \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}$$

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. $$r=\sin ^{5} \theta+8 \sin \theta \cos ^{3} \theta$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{i}$$

If \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j},\) find a vector orthogonal to \(\mathbf{v}\)

Find the angle, in degrees, between \(\mathbf{v}\) and \(\mathbf{w}.\) $$\mathbf{v}=2 \cos \frac{4 \pi}{3} \mathbf{i}+2 \sin \frac{4 \pi}{3} \mathbf{j}, \quad \mathbf{w}=3 \cos \frac{3 \pi}{2} \mathbf{i}+3 \sin \frac{3 \pi}{2} \mathbf{j}$$
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