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Chapter 7: Problem 29

Test for symmetry and then graph each polar equation. $$r^{2}=9 \cos 2 \theta$$

Short Answer

Expert verified
The curve of the given polar equation is symmetric about the origin and the graph will resemble an 'infinity' symbol or a 'Lemniscate'.

Step by step solution

01

Analyzing the Exercise

The given equation is of the form \(r^{2} = 9\cos(2\theta)\). It is clear that the equation is in polar coordinates and needs to be graphed. Before graphing the equation, it's important to verify if the curve is symmetrical.
02

Checking for symmetry about the Polar Axis

Replace \(\theta\) by \(-\theta\) in the equation. If the equation remains the same then it is symmetric about the polar axis or x-axis. But here, when we replace \(\theta\) by \(-\theta\) the equation changes to \(r^{2} = 9\cos (-2\theta)\), which is not the same as the original. Therefore, the curve is not symmetric about the polar axis.
03

Checking for symmetry about the Line 胃=蟺/2

We replace \(\theta\) by \(\pi - \theta\). If the resulting equation matches the original one, then the curve is symmetric about the line \(\theta = \pi/2\) or y-axis. On replacing \(\theta\) by \(\pi - \theta\), the equation becomes \(r^{2} = 9\cos(2\pi - 2\theta)\), which does not match the original equation. Hence the curve is not symmetric about the line 胃=蟺/2.
04

Checking for symmetry about the Origin

Replace \(r\) with \(-r\) and see if we get the same equation. This checks symmetry with respect to origin. It's because, replacing \(r\) with \(-r\) and getting the same equation back means that all points (r, 胃) and (-r, 胃) (which are diametrically opposite each other) are on the graph. The equation remains \(r^{2} = 9\cos(2\theta)\), even when \(r\) is replaced with \(-r\). Therefore, the curve is symmetric with respect to the origin.
05

Plotting the curve

Now that we've determined the symmetries prevalent in the graph, the next step is to plot the curve for \(r^{2} = 9\cos(2\theta)\). This is a Lemniscate, tha graph of which would be similar to an infinity symbol. Remember to make note that the equation is symmetric with respect to origin.

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Most popular questions from this chapter

In Exercises \(81-86,\) solve equation in the complex number system. Express solutions in polar and rectangular form. $$ x^{3}-(1+i \sqrt{3})=0 $$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The proof of the formula for the product of two complex numbers in polar form uses the sum formulas for cosines and sines that I studied in the previous chapter.

In Exercises \(81-86,\) solve equation in the complex number system. Express solutions in polar and rectangular form. $$ x^{6}+1=0 $$

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

Use a graphing utility to graph the polar equation. $$r=4 \sin 5 \theta$$
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