Recall that i = √-1 € C, and let 0 € R. Prove that for all integers n ≥ 0, we have (cos+ i sin 0) = cos(n) + i sin(ne). You cannot use the fact that eie = cos 0+isin, or similarly anything involving e or polar coordinates. You will find the compound angle formulas helpful: for all a, ß ER, we have cos a cos 3- sin a sin 3 = cos(a + B), sin a cos 3 + cos a sin 3 = sin(a+ß).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Recall that \( i = \sqrt{-1} \in \mathbb{C} \), and let \(\theta \in \mathbb{R}\). Prove that for all integers \( n \geq 0 \), we have

\[
(\cos \theta + i \sin \theta)^n = \cos(n \theta) + i \sin(n \theta).
\]

You cannot use the fact that \( e^{i \theta} = \cos \theta + i \sin \theta \), or similarly anything involving \( e \) or polar coordinates. You will find the ***compound angle formulas*** helpful: for all \(\alpha, \beta \in \mathbb{R}\), we have

\[
\cos \alpha \cos \beta - \sin \alpha \sin \beta = \cos(\alpha + \beta),
\]
\[
\sin \alpha \cos \beta + \cos \alpha \sin \beta = \sin(\alpha + \beta).
\]
Transcribed Image Text:Recall that \( i = \sqrt{-1} \in \mathbb{C} \), and let \(\theta \in \mathbb{R}\). Prove that for all integers \( n \geq 0 \), we have \[ (\cos \theta + i \sin \theta)^n = \cos(n \theta) + i \sin(n \theta). \] You cannot use the fact that \( e^{i \theta} = \cos \theta + i \sin \theta \), or similarly anything involving \( e \) or polar coordinates. You will find the ***compound angle formulas*** helpful: for all \(\alpha, \beta \in \mathbb{R}\), we have \[ \cos \alpha \cos \beta - \sin \alpha \sin \beta = \cos(\alpha + \beta), \] \[ \sin \alpha \cos \beta + \cos \alpha \sin \beta = \sin(\alpha + \beta). \]
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