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Chapter 6: Problem 76

Explain why the graph of \(r=\cos 2 \theta\) and the graph of \(r=2 \cos ^{2} \theta-1\) are identical.

Short Answer

Expert verified
The reason why the graph of \(r=\cos 2 \theta\) and the graph of \(r=2 \cos ^{2} \theta-1\) are identical is because they are mathematically identical, derived using the double-angle identity of cosine.

Step by step solution

01

Use the Double-Angle Identity

The first step is to use a double-angle identity of cos, which states that \(cos 2 \theta = 2cos^2\theta - 1\). This is a well-known trigonometric identity that can be used to simplify the equation.
02

Substitute the identity into the first equation

Once the identity is obtained, it should be substituted into the original equation. This means that \(r=\cos 2 \theta\) can be expressed as \(r=2 \cos ^{2}\theta - 1\).
03

Compare the two equations

With this new form of the equation, you can easily see that both equations are identical. This is sufficient to prove that they will yield the same graph on a plot between r and \(\theta\).

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