If the matrix `[{:(cos theta,sin theta,0),(sin theta,cos theta,0),(0,0,1):}]` is singular, then what is one of the values of `theta`?

Options

A. `(pi)/(4)`
B. `(pi)/(2)`
C. `pi`
D. 0

Explanation

To determine one of the values of \( \theta \) for which the given matrix

\[ A = \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

is singular, we need to find when its determinant is equal to zero.

### Step 1: Calculate the Determinant of the Matrix
The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

- \( a = \cos \theta \), \( b = \sin \theta \), \( c = 0 \)
- \( d = \sin \theta \), \( e = \cos \theta \), \( f = 0 \)
- \( g = 0 \), \( h = 0 \), \( i = 1 \)

Substituting these values into the determinant formula gives:

\[ \text{det}(A) = \cos \theta (\cos \theta \cdot 1 - 0 \cdot 0) - \sin \theta (\sin \theta \cdot 1 - 0 \cdot 0) + 0 \]

This simplifies to:

\[ \text{det}(A) = \cos^2 \theta - \sin^2 \theta \]

### Step 2: Set the Determinant Equal to Zero
Since the matrix is singular, we set the determinant equal to zero:

\[ \cos^2 \theta - \sin^2 \theta = 0 \]

### Step 3: Use the Pythagorean Identity
We can rewrite the equation as:

\[ \cos^2 \theta = \sin^2 \theta \]

This can be expressed using the identity:

\[ \cos^2 \theta - \sin^2 \theta = \cos 2\theta \]

Thus, we have:

\[ \cos 2\theta = 0 \]

### Step 4: Solve for \( 2\theta \)
The cosine function is zero at odd multiples of \( \frac{\pi}{2} \):

\[ 2\theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]

### Step 5: Solve for \( \theta \)
Dividing by 2 gives:

\[ \theta = \frac{\pi}{4} + \frac{n\pi}{2} \]

### Conclusion
One of the values of \( \theta \) is:

\[ \theta = \frac{\pi}{4} \]