Question

which angle of rotation is determined by the matrix below?\
\\(\

$$\begin{bmatrix}\\frac{1}{2} & -\\frac{\\sqrt{3}}{2}\\\\ \\frac{\\sqrt{3}}{2} & \\frac{1}{2}\\end{bmatrix}$$

\\)\
\\(\bigcirc\\) \\(30^\circ\\)\
\\(\bigcirc\\) \\(60^\circ\\)\
\\(\bigcirc\\) \\(120^\circ\\)\
\\(\bigcirc\\) \\(300^\circ\\)

Explanation:

Step1: Recall the rotation matrix formula

The rotation matrix is \(

$$\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$$

\)

Step2: Compare with the given matrix

Given matrix \(

$$\begin{bmatrix}\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\frac{\sqrt{3}}{2}&\frac{1}{2}\end{bmatrix}$$

\), so \(\cos\theta=\frac{1}{2}\) and \(\sin\theta=\frac{\sqrt{3}}{2}\)

Step3: Find the angle \(\theta\)

Since \(\cos\theta=\frac{1}{2}\) and \(\sin\theta=\frac{\sqrt{3}}{2}\), \(\theta = 60^{\circ}\)

Answer:

\(60^{\circ}\)