12. {cos 2x, cos²x, sin²x} on (-∞, ∞)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7a11c6-35df-4daf-855c-043e8f69fb75%2F568145bc-7aa6-4332-94d3-01a50b4bb7fb%2Ft1zrexg_processed.png&w=3840&q=75)
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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