Question

Graph the polar equations. $$r^{2}=9 \cos 2 \theta$$

   Graph the polar equations.
$$r^{2}=9 \cos 2 \theta$$

Precalculus

David Cohen,… 7th Edition

Chapter 10, Problem 20

Instant Answer

Solved by Expert Nick Johnson

05/16/2025

Step 1

We know that $x = r\cos\theta$ and $y = r\sin\theta$. So, $x^2 + y^2 = r^2$ and $\cos 2\theta = 1 - 2\sin^2\theta$. Now, we can substitute these expressions into the given polar equation: $$r^2 = 9\cos 2\theta$$ $$x^2 + y^2 = 9(1 - 2\sin^2\theta)$$ We also Show more…

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Graph the polar equations. $$r^{2}=9 \cos 2 \theta$$

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Key Concepts

Lemniscate

A lemniscate is a specific type of polar curve that has a figure-eight or infinity shape, typically arising from equations of the form r² = a² cos(2?) or r² = a² sin(2?). Understanding the lemniscate involves recognizing its key characteristics, such as its symmetry and the angular intervals over which the expression for r is non-negative and thus yields real-valued points.

Polar Coordinates

This concept involves representing points in the plane using a radius and angle rather than the traditional x and y coordinates. It forms the basis for understanding curves defined in terms of r and ?, where r indicates the distance from the origin and ? the angular displacement from a reference direction.

Graphing Polar Equations

Graphing in polar coordinates requires identifying the relationship between the radius r and the angle ?, determining the valid domain for the equation, and plotting the resulting points. It involves understanding how variations in ? affect r and recognizing patterns such as loops, petals, or other distinctive features of the curves.

Symmetry in Polar Graphs

Symmetry plays a crucial role in simplifying the graphing process of polar equations. By analyzing the equation for invariance under transformations (such as reflecting across the polar axis or the line ? = ?/2), one can reduce the workload by plotting only a portion of the curve and then applying the symmetry.

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