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Chapter 7: Problem 70

Use a graphing utility to graph the polar equation. $$r=\cos \frac{3}{2} \theta$$

Short Answer

Expert verified
The graphing process will be easier if the equation is simplified into a Cartesian format and then graphed using a proper graphing tool.

Step by step solution

01

Transformation of Polar Equation to Cartesian Equation

To transform the given equation \(r=\cos (\frac{3}{2} \theta)\) into Cartesian form, use the relations \[x=r\cos(\theta)\] and \[y=r\sin(\theta).\] Since \(r^2 = x^2 + y^2\) and \(\theta = \arctan\left(\frac{y}{x}\right)\), replace \(r\) and \(\theta\) by \(x\) and \(y\) in the equation.
02

Simplify Equation

After substitution we get the equation in Cartesian form. Then simplify the equation as much as possible.
03

Graphing the Equation

After getting a simplified Cartesian equation, use a graphing utility to sketch the graph. Note that some graphing utilities allow for graphing in polar coordinates directly, in that case, the transformation steps can be skipped.

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Most popular questions from this chapter

Explain how to plot a complex number in the complex plane. Provide an example with your explanation.

Solve and graph the solution set on a number line: $$ |2 x+3| \leq 13 $$

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. $$r=4 \sin \theta$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with a polar equation that failed the symmetry test with respect to \(\theta=\frac{\pi}{2},\) so my graph will not have this kind of symmetry.

The image of the Mandelbrot set in the section opener exhibits self- similarity: Magnified portions repeat much of the pattern of the whole structure, as well as new and unexpected patterns. Describe an object in nature that exhibits self-similarity.
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