If $
\begin{bmatrix}
\cos \frac{2 \pi}{7} & -\sin \frac{2 \pi}{7} \\
\sin \frac{2 \pi}{7} & \cos \frac{2 \pi}{7} \\
\end{bmatrix}^k
=
\begin{bmatrix}
1 & 0 \\
0 & 1 \\
\end{bmatrix}
$, then the least positive integral value of $k$ is?
Actually, I got no idea how to solve this.
I did trial and error method and got 7 as the answer.
how do i solve this?
Can you please offer your assistance? Thank you
Best Answer
We start with polar coordinates of a point $P=(r,\theta)$ vs its Cartesian form $(x,y)$. You know $x=r\cos\theta, y=r\sin\theta$. Now rotate the point $P$ by an angle of $\alpha$ with respect to origin so its angle with respect to $x$ axis becomes $\theta+\alpha$. Its new polar coordinates is $P'=(r,\theta+\alpha)$ and its new Cartesian coordinates $(x',y')$, where $x'=r\cos(\theta+\alpha), y'=r\sin(\theta+\alpha)$. Use trig identities to expand these
$x'=r(\cos \theta \cos \alpha -\sin \theta \sin \alpha)=x\cos \alpha -y \sin\alpha$, and
$ y'=r(\sin\theta\cos\alpha+\cos\theta\sin\alpha)=y\cos\alpha+x\sin\alpha$.
If you write these as a matrix equation you can finish your problem. $ \left[ \begin{array}{c} x' \\ y' \end{array} \right] = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{bmatrix} \times \left[ \begin{array}{c} x \\ y \end{array} \right] $
If you have several rotation $\alpha_1, \cdots, \alpha_n$ then the effect of multiplying all the related matrices is the same as adding the given angles. In particular if you have $n$ rotations by the same angle $\alpha$ the result will be same as one rotation of $n\alpha$. In short $ \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \\ \end{bmatrix}^n= \begin{bmatrix} \cos n\alpha & -\sin n\alpha \\ \sin n\alpha & \cos n\alpha \\ \end{bmatrix}$