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).
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
Section: Chapter Questions
Problem 1RQ
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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
\[
\cos \alpha \cos \beta - \sin \alpha \sin \beta = \cos(\alpha + \beta),
\]
\[
\sin \alpha \cos \beta + \cos \alpha \sin \beta = \sin(\alpha + \beta).
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F382d69cf-ffe2-43c0-99c4-21bedf550518%2F91b0c4f9-c083-4c40-bf75-9e54888ed8b0%2F9xgrmv_processed.png&w=3840&q=75)
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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