Question

Evaluate:$\left|\begin{array}{cc}\cos {30}^{\circ }& \sin {30}^{\circ }\\ \sin {105}^{\circ }& \cos {105}^{\circ }\end{array}\right|$

Answer 1

$\left|\begin{array}{cc}\cos {30}^{\circ }& \sin {30}^{\circ }\\ \sin {105}^{\circ }& \cos {105}^{\circ }\end{array}\right|$

$=\cos {30}^{\circ }\cos {105}^{\circ }-\sin {30}^{\circ }\sin {105}^{\circ }$

$=\cos ({30}^{\circ }+{105}^{\circ })$ using $\cos A\cos B-\sin A\sin B=\cos (A+B)$

$=\cos \left({135}^{\circ }\right)$

$=\cos ({90}^{\circ }+{45}^{\circ })$

$=-\sin {45}^{\circ }=-\frac{1}{\sqrt{2}}$