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12. {cos 2x, cos²x, sin²x} on (-∞, ∞)

12. {cos 2x, cos²x, sin²x} on (-∞, ∞)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**12.** \(\{ \cos 2x, \cos^2 x, \sin^2 x \}\) on \((-∞, ∞)\) This problem involves analyzing the set of functions \(\cos 2x\), \(\cos^2 x\), and \(\sin^2 x\) across the entire real number line. - **\(\cos 2x\):** This is a standard trigonometric function representing the cosine of twice the angle \(x\). The function has a period of \(\pi\), meaning it repeats every \(\pi\) units along the x-axis. - **\(\cos^2 x\):** This function is the square of the cosine function. It describes how the cosine of angle \(x\) behaves when squared, affected by its period of \(2\pi\). - **\(\sin^2 x\):** Similarly, this is the square of the sine function. Like \(\cos^2 x\), it also has a period of \(2\pi\). These functions are often analyzed together in contexts involving trigonometric identities or Fourier series on infinite intervals.
Transcribed Image Text:**12.** \(\{ \cos 2x, \cos^2 x, \sin^2 x \}\) on \((-∞, ∞)\) This problem involves analyzing the set of functions \(\cos 2x\), \(\cos^2 x\), and \(\sin^2 x\) across the entire real number line. - **\(\cos 2x\):** This is a standard trigonometric function representing the cosine of twice the angle \(x\). The function has a period of \(\pi\), meaning it repeats every \(\pi\) units along the x-axis. - **\(\cos^2 x\):** This function is the square of the cosine function. It describes how the cosine of angle \(x\) behaves when squared, affected by its period of \(2\pi\). - **\(\sin^2 x\):** Similarly, this is the square of the sine function. Like \(\cos^2 x\), it also has a period of \(2\pi\). These functions are often analyzed together in contexts involving trigonometric identities or Fourier series on infinite intervals.
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