Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Solution 2

We will follow the second hint. Let $v\in \mathbb {R} ^{2}$ be a vector that makes an angle of $30^{\circ }$ with the $x$-axis. Then the first reflection fixes $v$, so the composite ${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$ takes $v$ into ${\mathit {Ref}}_{50^{\circ }}(v)$, which is a vector that makes an angle of 40º with $v$, in the counterclock-wise direction. Therefore, ${\mathit {Ref}}_{50^{\circ }}\circ {\mathit {Ref}}_{30^{\circ }}$ is equal to rotation by 40º.