A description/proof of that 2 x 2 special orthogonal matrix can be expressed with sine and cosine of angle
Topics
About: matrix
The table of contents of this article
Starting Context
- The reader knows a definition of matrix.
Target Context
- The reader will have a description and a proof of the proposition that any 2 x 2 special orthogonal matrix can be expressed with the sine and the cosine of an angle.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description
Any 2 x 2 special orthogonal matrix, \(M\), can be expressed as \(\begin{pmatrix} sin \theta & - cos \theta \\ cos \theta & sin \theta\end{pmatrix}\) where \(\theta\) is an angle such that \(0 \leq \theta \lt 2\pi\).
2: Proof
Let \(M\) be \(\begin{pmatrix} a & c \\ b & d\end{pmatrix}\). \(M^t M = \begin{pmatrix} a^2 + b^2 & ac + bd \\ ac + bd & c^2 + d^2\end{pmatrix} = I\) and \(det M = ad - bc = 1\). \(a = sin \theta, b = cos \theta\) for a \(\theta\) such that \(0 \leq \theta \lt 2\pi\). \(c = - cos \theta', d = sin \theta'\) for a \(\theta'\) such that \(0 \leq \theta' \lt 2\pi\). \(- sin \theta cos \theta' + cos \theta sin \theta' = 0\), \(tan \theta = tan \theta'\). \(\theta' = \theta\) or \(\theta' = \theta + \pi\) or \(\theta' = \theta - \pi\). \(sin \theta sin \theta' + cos \theta cos \theta' = 1\). So, \(\theta' = \theta\).
