# Evaluate the determinant: $\begin{vmatrix} cos\theta &-sin\theta \\ sin\theta & cos\theta \end{vmatrix}$

Toolbox:
• For a given determinant A of order 2 $\begin{vmatrix}a_{11}& a_{12}\\a_{21} & a_{22}\end{vmatrix}$
• To evaluate the value of the given determinants ,let us multiply the elements $a_{11}$ and $a_{22}$ and then subtract $a_{21}\times a_{12}$.

Given (i) $A=\begin{vmatrix}cos\theta &-sin\theta\\sin\theta & cos\theta\end{vmatrix}$

To evaluate the value of the given determinants ,let us multiply the elements $a_{11}$ and $a_{22}$ and then subtract $a_{21}\times a_{12}$.

$\mid A\mid=cos \theta\times cos\theta-(sin\theta)sin\theta$.

$\qquad=cos^2\theta+sin^2\theta$

But $cos^2\theta+sin^2\theta$=1.

Therefore $\mid A\mid=1.$

edited Feb 24, 2013