Let \(F(x) = \begin{bmatrix}\cos x&-\sin x&0\\\sin x&\cos x&0\\0&0&1\end{bmatrix}\). The element in the first row, second column of the matrix \(F\left(\displaystyle\frac{\pi}{12}\right)F\left(\displaystyle\frac{\pi}{4}\right)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-\displaystyle\frac{\sqrt{3}}{2}\)
Let us multiply \(F(x)\) and \(F(y)\) in general. The element in the first row, second column is row 1 of \(F(x)\) times column 2 of \(F(y)\):
\(\cos x(-\sin y) + (-\sin x)\cos y + 0(0)\)
\(= -(\sin x\cos y + \cos x\sin y)\)
\(= -\sin(x + y)\)
The other elements work out in the same way, so \(F(x)F(y) = F(x + y)\). Here
\( \displaystyle x + y = \frac{\pi}{12} + \frac{\pi}{4} = \frac{\pi}{3} \)
So the required element is
\( \displaystyle -\sin\frac{\pi}{3} = -\frac{\sqrt{3}}{2} \)