The roots of the equation
`|(3x^(2),x^(2) + x cos theta + cos^(2) theta ,x^(2) + x sin theta + sin^(2) theta),(x^(2) + x cos theta + cos^(2) theta,3 cos^(2) theta,1 + (sin 2 theta)/(2)),(x^(2) + x sin theta + sin^(2) theta,1 + (sin 2 theta)/(2),3 sin^(2) theta)| = 0`

Options

A. `sin theta, cos theta`
B. `sin^(2) theta, cos^(2) theta`
C. `sin theta, cos^(2) theta`
D. `sin^(2) theta, cos theta`

Answer

To solve the given determinant equation, we will follow these steps:

### Step 1: Write the determinant
We start with the determinant given in the problem:
\[ D = \begin{vmatrix} 3x^2 & x^2 + x \cos \theta + \cos^2 \theta & x^2 + x \sin \theta + \sin^2 \theta \\ x^2 + x \cos \theta + \cos^2 \theta & 3 \cos^2 \theta & 1 + \frac{\sin 2\theta}{2} \\ x^2 + x \sin \theta + \sin^2 \theta & 1 + \frac{\sin 2\theta}{2} & 3 \sin^2 \theta \end{vmatrix} \]

### Step 2: Factor out common terms
We can factor out \(3\) from the first row:
\[ D = 3 \begin{vmatrix} x^2 & \frac{1}{3}(x^2 + x \cos \theta + \cos^2 \theta) & \frac{1}{3}(x^2 + x \sin \theta + \sin^2 \theta) \\ \frac{1}{3}(x^2 + x \cos \theta + \cos^2 \theta) & \cos^2 \theta & \frac{1}{3}(1 + \frac{\sin 2\theta}{2}) \\ \frac{1}{3}(x^2 + x \sin \theta + \sin^2 \theta) & \frac{1}{3}(1 + \frac{\sin 2\theta}{2}) & \sin^2 \theta \end{vmatrix} \]

### Step 3: Simplify the determinant
Next, we can rewrite the determinant as a product of two smaller determinants:
\[ D = \begin{vmatrix} x^2 & x & 1 \\ \cos^2 \theta & \cos \theta & 1 \\ \sin^2 \theta & \sin \theta & 1 \end{vmatrix} \]

### Step 4: Perform row operations
We perform row operations to simplify the determinant:
- \(R_2 \leftarrow R_2 - R_1\)
- \(R_3 \leftarrow R_3 - R_1\)

This gives us:
\[ D = \begin{vmatrix} x^2 & x & 1 \\ \cos^2 \theta - x^2 & \cos \theta - x & 0 \\ \sin^2 \theta - x^2 & \sin \theta - x & 0 \end{vmatrix} \]

### Step 5: Expand the determinant
Now we can expand the determinant along the last column:
\[ D = (0) - \begin{vmatrix} x^2 & x \\ \cos^2 \theta - x^2 & \cos \theta - x \end{vmatrix} + \begin{vmatrix} x^2 & x \\ \sin^2 \theta - x^2 & \sin \theta - x \end{vmatrix} \]

### Step 6: Solve the determinant
Now we solve the determinants:
\[ D = (x^2)(\cos \theta - x) - (x)(\cos^2 \theta - x^2) + (x^2)(\sin \theta - x) - (x)(\sin^2 \theta - x^2) \]

### Step 7: Set the determinant to zero
Set the determinant equal to zero:
\[ D = 0 \]

### Step 8: Find the roots
From the determinant, we can find the roots:
1. \(x = \cos \theta\)
2. \(x = \sin \theta\)

### Conclusion
The roots of the equation are:
\[ x = \cos \theta \quad \text{and} \quad x = \sin \theta \]

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