# Science:Math Exam Resources/Courses/MATH152/April 2022/Question A18/Solution 1

Let ${\textstyle L1}$ be the line that makes an angle of ${\textstyle 30^{\circ }}$ with the positive ${\textstyle x_{1}}$-axis, and ${\textstyle L2}$ be the line that makes an angle of ${\textstyle 50^{\circ }}$ with the positive ${\textstyle x_{1}}$-axis. We sketch the sequence of reflections below, keeping the original orientation of the axes in the background, labelled as ${\displaystyle x_{1}(1)}$, ${\displaystyle x_{2}(1)}$. In the sketch, the image of the ${\displaystyle x_{1}(1)}$-, ${\displaystyle x_{2}(1)}$-axes after ${\displaystyle {\mathit {Ref}}_{30^{\circ }}}$ is labelled as ${\displaystyle x_{1}(2)}$, ${\displaystyle x_{2}(2)}$. The reflection ${\displaystyle {\mathit {Ref}}_{50^{\circ }}}$ is then applied to the ${\displaystyle x_{1}(2)}$-, ${\displaystyle x_{2}(2)}$-axes, resulting in the ${\displaystyle x_{1}(3)}$-, ${\displaystyle x_{2}(3)}$-axes.

Math 152 2022 Q18 solution figure

After the reflection about ${\textstyle L2}$, we now look at the angle between the axes ${\textstyle x_{1}(1)}$ and ${\textstyle x_{1}(3)}$, this is ${\textstyle \theta =50^{\circ }-10^{\circ }=40^{\circ }}$ (shown in purple). The ${\textstyle x_{1}}$- and ${\textstyle x_{2}}$-axes are now ${\textstyle \theta =40^{\circ }}$ off from their original orientation (taking counterclockwise rotation to be positive).