Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Let \( R_{\alpha} : \mathbb{R}^2 \to \mathbb{R}^2 \) be rotation by \( \alpha \) radians counterclockwise. What is the standard matrix of \( R_{\alpha} \)? Prove the addition formulas for \(\cos(\alpha + \beta)\) and \(\sin(\alpha + \beta)\) by considering the composite linear map \( R_{\beta} \circ R_{\alpha} \).
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Transcribed Image Text:Let \( R_{\alpha} : \mathbb{R}^2 \to \mathbb{R}^2 \) be rotation by \( \alpha \) radians counterclockwise. What is the standard matrix of \( R_{\alpha} \)? Prove the addition formulas for \(\cos(\alpha + \beta)\) and \(\sin(\alpha + \beta)\) by considering the composite linear map \( R_{\beta} \circ R_{\alpha} \).
Expert Solution
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Step 1: Defining the standard matrix

Given Rα:ℝ2→ℝ2

Rα=cosα-sinαsinαcosα

Therefore the standard matrix of Rα is cosα-sinαsinαcosα

 

 

Advanced Math homework question answer, step 1, image 1

 

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