If the system of linear equations
`{:((cos theta)x + (sin theta) y + cos theta=0),((sin theta)x+(cos theta)y + sin theta=0),((cos theta)x + (sin theta)y -cos theta=0):}`
is consistent , then the number of possible values of `theta, theta in [0,2pi]` is :

Answer

To determine the number of possible values of \( \theta \) for which the given system of linear equations is consistent, we will analyze the determinant of the coefficient matrix formed by the equations.

The system of equations is:
1. \( (\cos \theta)x + (\sin \theta)y + \cos \theta = 0 \)
2. \( (\sin \theta)x + (\cos \theta)y + \sin \theta = 0 \)
3. \( (\cos \theta)x + (\sin \theta)y - \cos \theta = 0 \)

We can rewrite these equations in matrix form as follows:

\[ \begin{bmatrix} \cos \theta & \sin \theta & 1 \\ \sin \theta & \cos \theta & 1 \\ \cos \theta & \sin \theta & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

To check for consistency, we need to find the determinant of the coefficient matrix. The determinant \( D \) of the matrix is given by:

\[ D = \begin{vmatrix} \cos \theta & \sin \theta & 1 \\ \sin \theta & \cos \theta & 1 \\ \cos \theta & \sin \theta & -1 \end{vmatrix} \]

We can calculate this determinant using the formula for the determinant of a 3x3 matrix:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Substituting the values from our matrix:

\[ D = \cos \theta \left( \cos \theta \cdot (-1) - \sin \theta \cdot 1 \right) - \sin \theta \left( \sin \theta \cdot (-1) - \cos \theta \cdot 1 \right) + 1 \left( \sin \theta \cdot \sin \theta - \cos \theta \cdot \cos \theta \right) \]

Calculating each term:

1. First term: \( \cos \theta (-\cos \theta - \sin \theta) = -\cos^2 \theta - \cos \theta \sin \theta \)
2. Second term: \( -\sin \theta (-\sin \theta - \cos \theta) = \sin^2 \theta + \sin \theta \cos \theta \)
3. Third term: \( \sin^2 \theta - \cos^2 \theta \)

Combining these terms:

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

Simplifying:

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

This can be factored as:

\[ D = 2(\sin^2 \theta - \cos^2 \theta) = 2(-\cos 2\theta) \]

For the system to be consistent, the determinant must be zero:

\[ 2(-\cos 2\theta) = 0 \implies \cos 2\theta = 0 \]

The solutions to \( \cos 2\theta = 0 \) occur at:

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

Thus,

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

Now we need to find the values of \( \theta \) in the interval \( [0, 2\pi] \):

1. For \( n = 0 \): \( \theta = \frac{\pi}{4} \)
2. For \( n = 1 \): \( \theta = \frac{3\pi}{4} \)
3. For \( n = 2 \): \( \theta = \frac{5\pi}{4} \)
4. For \( n = 3 \): \( \theta = \frac{7\pi}{4} \)

These values are all within the interval \( [0, 2\pi] \).

Thus, the number of possible values of \( \theta \) is \( 4 \).

**Final Answer:** The number of possible values of \( \theta \) in the interval \( [0, 2\pi] \) is \( 4 \).

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