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Chapter 7: Problem 27

Test for symmetry and then graph each polar equation. $$r=4 \sin 3 \theta$$

Short Answer

Expert verified
The given equation is symmetric with respect to the origin. It will result in a rose with three petals, each of length 4 units.

Step by step solution

01

Symmetry With Respect To The Polar Axis

Substitute '(-胃)' for '胃' in the given equation and simplify. Given, \(r = 4\sin(3\theta)\).Let's substitute \(-\theta\) for \(\theta\).If the result is equivalent to the original equation, then the graph is symmetric with respect to the polar axis. Here, \(r = 4\sin(3(-\theta)) = 4\sin(-3\theta)\). Since \(\sin(-\theta) = - \sin(\theta)\), the equation simplifies to \(r = -4\sin(3\theta)\) which is not equivalent to the original equation, so the graph is not symmetric about the polar axis.
02

Symmetry With Respect To The Pole

Substitute '(-r)' for 'r' in the given equation and simplify.So, substitute \(-r = 4\sin(3\theta)\) for \(r\). If the result is equivalent to the original equation, then the graph is symmetric with respect to the pole. But in this case, \(-r = 4\sin(3\theta)\) is not equivalent to the original equation, which means the graph is not symmetric about the pole.
03

Symmetry With Respect To The Origin

Substitute '(-胃)' for '胃' and '(-r)' for 'r' in the given equation.After substitution, we get \(-r = 4\sin(3(-\theta)) = -4\sin(3\theta)\). But \(\sin(-\theta) = - \sin(\theta)\). Therefore, the equation becomes \(-r = -4\sin(3\theta)\) or \(r = 4\sin(3\theta)\), which is the same as the original equation. So, the graph is symmetric about the origin.
04

Graph the Polar Equation

To graph this equation, select values for \(\theta\) between 0 and \(2\pi\) and then find corresponding 'r' values. Plot these points. The equation will create a rose with three leaves because of the '3' in \(\sin(3\theta)\), which indicates the number of leaves on the rose. The '4' in 'r=4sin3胃' represents the length of the petals.

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Most popular questions from this chapter

Find the smallest interval for \(\theta\) starting with \(\theta \min =0\) so that your graphing utility graphs the given polar equation exactly once without retracing any portion of it. $$r=4 \sin \theta$$

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 \(a+b i .\) $$ (1+i)(2+2 i) $$

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

In calculus, it can be shown that $$e^{i \theta}=\cos \theta+i \sin \theta$$ In Exercises \(87-90,\) use this result to plot each complex number. $$ -e^{-\pi i} $$

Use a graphing utility to graph the polar equation. $$r=2+4 \sin \theta$$
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