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Chapter 7: Problem 66

Sketch the graph of each polar equation. $$r^{2}=4 \sin 2 \theta \text { (lemniscate) }$$

Short Answer

Expert verified
Plot key points at particular angles and connect them to form a lemniscate.

Step by step solution

01

- Understand the Polar Equation

The given polar equation is: \[ r^2 = 4 \sin(2\theta) \]. This equation defines a lemniscate, which is a type of figure-eight or infinity symbol shape.
02

- Analyze the Symmetry

The equation \( r^2 = 4 \sin(2\theta) \) exhibits symmetry about the origin. This means if \( (r, \theta) \) is on the graph, then \( (-r, \theta + \pi) \) is also on the graph.
03

- Identify Key Points

Determine key values of \(\theta\) where notable points occur. For instance, consider \(\theta = \frac{\pi}{4}\): \[ r^2 = 4 \sin\left(2 \times \frac{\pi}{4} \right) = 4 \sin\left(\frac{\pi}{2}\right) = 4 \times 1 = 4 \] So, \( r = \pm 2 \). Similar points can be found for other values of \(\theta\).
04

- Plot the Key Points

Plot points calculated in Step 3 on the polar coordinate plane. Some key points are: When \(\theta = \frac{\pi}{4}, r = \pm 2\).When \(\theta = \frac{3\pi}{4}, r = \pm \sqrt{2}\) (same calculation approach).(Repeat similar calculations for other angles to get more points).
05

- Sketch the Shape

Using the points plotted, connect them to form the lemniscate shape. The graph should resemble a figure eight centered at the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

lemniscate
The term 'lemniscate' refers to a curve that looks like a figure-eight or an infinity symbol. In this exercise, the given equation is \( r^2 = 4 \sin(2 \theta) \), which defines a lemniscate. This equation is special because it is a polar equation, specifically designed for polar coordinates.

Lemniscates are unique because they have two loops. Imagine a graph shaped like an infinity sign, where the intersection occurs at the origin (0, 0).

One crucial aspect of a lemniscate is that it is symmetric about the origin, which means it looks the same if flipped across the origin. Understanding this helps in plotting its points accurately.
polar coordinates
Polar coordinates are a way of representing points on a plane through a distance from a fixed point (the origin) and an angle from a fixed direction (usually the positive x-axis). They are written as \( (r, \theta) \), where
  • \
symmetry analysis
The polar equation \( r^2 = 4 \sin(2 \theta) \) shows symmetry about the origin. This means if you have a point \( (r, \theta) \), then the point with coordinates \( (-r, \theta + \pi) \) will also lie on the graph.

Symmetry analysis is essential because it simplifies plotting the graph. If one side of the graph is known, you can mirror it to obtain the other side.

In the case of a lemniscate, this symmetry ensures the two loops are exact mirror images of each other. For example, if \( (2, \frac{\pi}{4}) \) is a point, then \( (-2, \frac{5\pi}{4}) \) will also be a point on the lemniscate.
key points plotting
When plotting a polar graph, it's crucial to determine and plot key points. For the lemniscate \( r^2 = 4 \sin(2 \theta) \), start by identifying values of \( \theta \) that make the equation simple to solve.
For instance, take \( \theta = \frac{\pi}{4} \):
\

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