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 solve the problem, we need to determine the values of \(\theta\) for which the given system of linear equations is consistent. The equations are:

1. \((\cos \theta)x + (\sin \theta)y + \cos \theta = 0\)
2. \((\sin \theta)x + (\cos \theta)y + \sin \theta = 0\)
3. \((\cos \theta)x + (\sin \theta)y - \cos \theta = 0\)

### Step 1: Write the system in matrix form

We can represent the system of equations in matrix form as follows:

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

### Step 2: Find the determinant of the coefficient matrix

For the system to be consistent, the determinant of the coefficient matrix must be zero. The determinant \(D\) of the matrix is given by:

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

### Step 3: Calculate the determinant

We can use the formula for the determinant of a \(3 \times 3\) 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: \(-\cos^2 \theta - \sin \theta\)
2. Second term: \(\sin^2 \theta + \sin \theta \cos \theta\)
3. Third term: \(\sin^2 \theta - \cos^2 \theta\)

Combining these gives:

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

### Step 4: Simplify the determinant

Combining like terms:

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

Setting \(D = 0\) for consistency:

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

### Step 5: Factor the equation

Factoring out common terms, we can rewrite it as:

\[ 2(\sin^2 \theta - \cos^2 \theta) + \sin \theta (\cos \theta - 1) = 0 \]

### Step 6: Solve for \(\theta\)

This gives us two cases to consider:

1. \(2(\sin^2 \theta - \cos^2 \theta) = 0\) leads to \(\sin^2 \theta = \cos^2 \theta\) or \(\tan^2 \theta = 1\), giving \(\theta = \frac{\pi}{4}, \frac{5\pi}{4}\).
2. \(\sin \theta (\cos \theta - 1) = 0\) gives \(\sin \theta = 0\) or \(\cos \theta = 1\), leading to \(\theta = 0, \pi, 2\pi\).

### Step 7: Count the unique solutions

The possible values of \(\theta\) in the interval \([0, 2\pi]\) are:

- From \(\tan^2 \theta = 1\): \(\frac{\pi}{4}, \frac{5\pi}{4}\)
- From \(\sin \theta = 0\): \(0, \pi, 2\pi\)

Thus, the unique values of \(\theta\) are:

- \(0\)
- \(\frac{\pi}{4}\)
- \(\pi\)
- \(\frac{5\pi}{4}\)
- \(2\pi\)

### Conclusion

The total number of unique values of \(\theta\) in the interval \([0, 2\pi]\) is **5**.