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Chapter 6: Problem 70

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

Short Answer

Expert verified
The graph of the polar equation \( r = \cos \left( \frac{3}{2} \theta \right) \) is a rose curve with three petals.

Step by step solution

01

Understand the Polar Coordinates System

In a polar coordinate system, each point in the plane is determined by an angle and a distance. The angle measurement \( \theta \) is positive in the counter-clockwise direction. It's necessary to understand this system to graph the given polar equation.
02

Identify the Form of the Equation

The polar equation \( r = \cos \left( \frac{3}{2} \theta \right) \) is a cosine function. In general, cosine functions in polar coordinates create a circle or circle-like shape, depending on the coefficient of \( \theta \). Here since the coefficient of \( \theta \) is \( \frac{3}{2} \), it will create a special kind of shape called a rose curve.
03

Plot the Polar Equation using a Graphing Utility

To visualize the function, use a graphing utility that accepts polar coordinates. Enter \( r = \cos \left( \frac{3}{2} \theta \right) \) into the graphing utility and sketch the resultant curve. It's important to ensure that the graph extends over an interval of at least \( 2 \pi \) for \( \theta \). For this equation, the graph should show a rose curve with three petals.

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Most popular questions from this chapter

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+10 \mathbf{j}$$

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}, \quad \mathbf{w}=-6 \mathbf{j}$$

Find the work done in pushing a car along a level road from point \(A\) to point \(B, 80\) feet from \(A,\) while exerting a constant force of 95 pounds. Round to the nearest foot-pound.

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=2 \mathbf{i}-2 \mathbf{j}, \quad \mathbf{w}=-\mathbf{i}+\mathbf{j}$$

Graph \(r_{1}\) and \(r_{2}\) in the same polar coordinate system. What is the relationship between the two graphs? $$r_{1}=2 \sin 3 \theta, r_{2}=2 \sin 3\left(\theta+\frac{\pi}{6}\right)$$
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