AIMS Lie Groups: Fall-2022
Practice Exercises 1 : Due 01 Mon 5 Dec

1. Euler's Formula II S0 4481S
   1. Look up or derive the Taylor series expansions for \(\cos\theta\), \(\sin\theta\), and \(e^x\).
   2. Show that \(e^{i\theta}=\cos\theta+i\sin\theta\).
   3. Verify that \(e^{2i\theta}=(e^{i\theta})^2\) using trigonometric identities, that is, show that \[ \cos2\theta+i\sin2\theta = (\cos\theta+i\sin\theta)^2 \]
2. The Rotation Group $SO(2)$ S0 4481S Consider rotation matrices of the form \[ M(\theta) = \begin{pmatrix} \cos\theta&-\sin\theta\\\sin\theta&\cos\theta \end{pmatrix} \]
   1. Show that
      1. \(M(0)=1\)
      2. \(M(-\theta)=M(\theta)^{-1}\)
      3. \(M(\alpha+\beta)=M(\alpha)M(\beta)\)
   2. Compute \(A=\dot{M}=\frac{dM}{d\theta}\Big|_{\theta=0}\).
   3. Show that \(M(\theta)=\exp(A\theta)\).