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Chapter 7: Problem 91

Determine whether each statement makes sense or does not make sense, and explain your reasoning. There are no points on my graph of \(r^{2}=9 \cos 2 \theta\) for which \(\frac{\pi}{4}<\theta<\frac{3 \pi}{4}\)

Short Answer

Expert verified
The statement makes sense because there are indeed no points on the graph of \(r^{2}=9 \cos 2 \theta\) for which \(\frac{\pi}{4}<\theta<\frac{3 \pi}{4}\). This is due to the cosine function in the given equation providing negative values in this range of \(\theta\), yet the term \(r^{2}\) cannot be negative.

Step by step solution

01

Understand the Graph

The graph of \(r^{2}=9 \cos 2 \theta\) is a lima莽on, which is a type of polar plot. The angular coefficient 2 indicates a faster variation of the radian than normal, meaning this lima莽on will have 'corners' or loops.
02

Understand the Range

For polar coordinates, \(\theta\) is the angle from the positive x-axis, and ranges from \(-\pi\) to \(\pi\) or from 0 to \(2\pi\). The given range of \(\frac{\pi}{4}<\theta<\frac{3 \pi}{4}\) corresponds to the second quadrant of the coordinate plane.
03

Determine the absence of points

In the equation \(r^{2}=9 \cos 2 \theta\), we can see that 'r' depends on the cosine function, which will be negative in the second quadrant, since \(\cos(2\theta)\) in the given range will fall into the third and fourth quadrant in unit circle, where cosine is either negative or zero. But, \(r^{2}\) can't be negative implying that there are no points in the given range of \(\theta\).

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Most popular questions from this chapter

Exercises \(99-101\) will help you prepare for the material covered in the next section. Refer to Section 1.4 if you need to review the basics of complex numbers. In each exercise, perform the indicated operation and write the result in the standard form \(a+b i .\) $$ (1+i)(2+2 i) $$

How do you determine the absolute value of a complex number?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I multiplied two complex numbers in polar form by first multiplying the moduli and then multiplying the arguments.

Explaining the Concepts What is the graph of a polar equation?

Use a graphing utility to graph the polar equation. $$r=\frac{3}{\sin \theta}$$
See all solutions

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