The value of `|{:(sin A ,cos A ),(- sin B, cos B):}|`, when A `= 54^(@) , B = 36^(@)` is a) 0 b) 1 c) -1 d) 2

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem, we need to evaluate the determinant of the matrix formed by the vectors \((\sin A, \cos A)\) and \((- \sin B, \cos B)\) when \(A = 54^\circ\) and \(B = 36^\circ\).

The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by the formula:

\[ |M| = ad - bc \]

In our case, the matrix is:

\[ \begin{pmatrix} \sin A & \cos A \\ -\sin B & \cos B \end{pmatrix} \]

Substituting \(A = 54^\circ\) and \(B = 36^\circ\), we have:

\[ \begin{pmatrix} \sin 54^\circ & \cos 54^\circ \\ -\sin 36^\circ & \cos 36^\circ \end{pmatrix} \]

Now, we can calculate the determinant:

\[ |M| = (\sin 54^\circ \cdot \cos 36^\circ) - (\cos 54^\circ \cdot (-\sin 36^\circ)) \]

This simplifies to:

\[ |M| = \sin 54^\circ \cdot \cos 36^\circ + \cos 54^\circ \cdot \sin 36^\circ \]

Using the sine addition formula, we recognize that:

\[ \sin(A + B) = \sin A \cdot \cos B + \cos A \cdot \sin B \]

Thus, we can rewrite our expression as:

\[ |M| = \sin(54^\circ + 36^\circ) \]

Calculating \(54^\circ + 36^\circ\):

\[ 54^\circ + 36^\circ = 90^\circ \]

Now, we know that:

\[ \sin 90^\circ = 1 \]

Therefore, the value of the determinant is:

\[ |M| = 1 \]

Thus, the answer is:

**b) 1**