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Chapter 7: Problem 71

Use a graphing utility to graph the polar equation. $$r=\cos \frac{5}{2} \theta$$

Short Answer

Expert verified
The detailed solution requires plotting the polar equation on a polar grid. As the equation doesn't easily convert to a Cartesian equation, tabulate \(\theta\) and \(r\) values to guide the plotting. The polar equation \(r = \cos \frac{5}{2}\theta\) creates a delicate flower-like structure with five petals, given the \(\frac{5}{2}\) multiplier.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are a type of coordinate system where each point in the space is defined by its distance \(r\) from the origin and its angle \(\theta\) from the positive horizontal axis. They are used when the relationship between two points is better expressed using their difference in angle and distance, instead of their difference in position within a Cartesian plane.
02

Convert the Polar Equation Into Cartesian Form

To graph a polar equation, you can first convert it into its Cartesian form. This can be done by substituting \(r = \sqrt{x^2 + y^2}\), \(x = r\cos \theta\), and \(y = r\sin \theta\) into the equation. However, the equation \(r = \cos \frac{5}{2} \theta\) doesn't easily convert into a Cartesian equation, so pick several values for \(\theta\) and calculate corresponding \(r\) values instead.
03

Create a Table of Values

Create a table with values of \(\theta\) and corresponding values of \(r\). This step requires you to input the \(\theta\) values into the given polar equation to get the \(r\) values. The table will guide you in plotting the graph of the polar equation.
04

Plot the Points Using Polar Coordinates

Use the table of values (obtained in Step 3) to plot points onto the polar graph by marking a point for each (\(r, \theta\)) pair.
05

Draw the Graph

After plotting all needed points, join them smoothly to draw the graph of the polar equation. The graph's shape will depend on your values in the table.

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Most popular questions from this chapter

Exercises \(119-121\) will help you prepare for the material covered in the next section. Find the obtuse angle \(\theta,\) rounded to the nearest tenth of a degree, satisfying. $$ \cos \theta=\frac{3(-1)+(-2)(4)}{| \mathbf{v}\|\mathbf{w}\|} $$ where \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{w}=-\mathbf{i}+4 \mathbf{j}\)

In Exercises \(81-86,\) solve equation in the complex number system. Express solutions in polar and rectangular form. $$ x^{5}-32 i=0 $$

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You want to fly your small plane due north, but there is a 75-kilometer wind blowing from west to east. a. Find the direction angle for where you should head the plane if your speed relative to the ground is 310 kilometers per hour. b. If you increase your airspeed, should the direction angle in part (a) increase or decrease? Explain your answer.

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.
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