Chapter 3.4, Problem 15E

a. Use trigonometric identities and mathematical induction to prove that

${\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]}^n=\left[\begin{array}{cc}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{array}\right]$

for all integers $n$ (positive, zero, negative). Hence conclude that for a constant $\theta$,

the set

$H=\left\{\left[\begin{array}{cc}\cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{array}\right]|n\in \mathbb{Z}\right\}$

is cyclic subgroup of the group of all invertible matrices in $M_2(ℝ)$.

b. Evaluate each element of $H$ for $\theta ={90}^{\circ }$.

c. Evaluate each element of $H$ for $\theta ={120}^{\circ }$.