Question

Sketch a graph of the polar equation. $$ r^{2}=\cos 2 \theta $$

   Sketch a graph of the polar equation.
$$
r^{2}=\cos 2 \theta
$$

Algebra and Trigonometry

Algebra and Trigonometry

James Stewart,… 3rd Edition

Chapter 9, Problem 37

Instant Answer

Solved by Expert Charles Carter

04/14/2020

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The equation $r^{2}=\cos 2 \theta$ is in the form of a lemniscate, which is a figure-eight shaped curve. Show more…

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Sketch a graph of the polar equation. $$ r^{2}=\cos 2 \theta $$

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Key Concepts

Trigonometric Functions in Polar Equations

Trigonometric functions, especially those involving multiples of the angle (such as double angles), are common in polar equations. Their periodic nature and range directly affect the domain and overall shape of the graph, making it necessary to analyze these functions to determine where the curve exists and how it behaves.

Lemniscate

A lemniscate is a type of curve that typically resembles a figure-eight and can be expressed in polar form with equations involving squared radius terms and double-angle trigonometric functions. Understanding the general shape, symmetry, and the conditions under which the radius is defined is essential when interpreting or sketching such curves.

Graphing Polar Equations

Graphing polar equations involves plotting the relationship between the radius and the angle, which often requires determining the valid range of angles for which the radius is defined and understanding how the radius changes as the angle varies. This process may also involve identifying features like intercepts, maximum and minimum radii, and regions of positive and negative values.

Polar Coordinates

Polar coordinates are a method for representing points in the plane using a radius and an angle instead of the traditional x and y coordinates. This system is particularly useful for describing curves that have inherent circular or rotational symmetry, and it simplifies the representation of certain types of curves that would be more complicated in Cartesian coordinates.

Symmetry in Polar Graphs

Symmetry plays an important role in simplifying the sketching of polar curves. Many polar equations exhibit symmetry about the polar axis, the line corresponding to a specific angle, or the origin. Recognizing these symmetries can allow one to plot only a segment of the curve and then reflect it to obtain the full graph.

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