To solve the problem, we need to evaluate the determinant of the given matrix and set it equal to zero. The matrix is:
\[
\begin{vmatrix}
\cos \theta & -1 & 1 \\
\cos 2\theta & 4 & 3 \\
2 & 7 & 7
\end{vmatrix}
= 0
\]
### Step 1: Calculate the Determinant
Using the determinant formula for a 3x3 matrix:
\[
D = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
where \( a, b, c \) are the elements of the first row, \( d, e, f \) are the elements of the second row, and \( g, h, i \) are the elements of the third row, we can substitute the values:
\[
D = \cos \theta (4 \cdot 7 - 3 \cdot 7) - (-1)(\cos 2\theta \cdot 7 - 3 \cdot 2) + 1(\cos 2\theta \cdot 7 - 4 \cdot 2)
\]
Calculating each term:
1. \( 4 \cdot 7 - 3 \cdot 7 = 28 - 21 = 7 \)
2. \( \cos 2\theta \cdot 7 - 3 \cdot 2 = 7 \cos 2\theta - 6 \)
3. \( \cos 2\theta \cdot 7 - 4 \cdot 2 = 7 \cos 2\theta - 8 \)
Putting it all together:
\[
D = \cos \theta \cdot 7 + (7 \cos 2\theta - 6) + (7 \cos 2\theta - 8)
\]
This simplifies to:
\[
D = 7 \cos \theta + 14 \cos 2\theta - 14
\]
### Step 2: Set the Determinant to Zero
Now we set the determinant equal to zero:
\[
7 \cos \theta + 14 \cos 2\theta - 14 = 0
\]
Dividing the entire equation by 7 gives:
\[
\cos \theta + 2 \cos 2\theta - 2 = 0
\]
### Step 3: Substitute for \(\cos 2\theta\)
Using the double angle identity, \( \cos 2\theta = 2 \cos^2 \theta - 1 \):
\[
\cos \theta + 2(2 \cos^2 \theta - 1) - 2 = 0
\]
This simplifies to:
\[
\cos \theta + 4 \cos^2 \theta - 2 - 2 = 0
\]
Which further simplifies to:
\[
4 \cos^2 \theta + \cos \theta - 4 = 0
\]
### Step 4: Solve the Quadratic Equation
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = 1, c = -4 \):
\[
\cos \theta = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 4 \cdot (-4)}}{2 \cdot 4}
\]
Calculating the discriminant:
\[
1 + 64 = 65
\]
Thus, we have:
\[
\cos \theta = \frac{-1 \pm \sqrt{65}}{8}
\]
### Step 5: Determine Valid Values of \(\theta\)
Calculating the two possible values:
1. \( \cos \theta = \frac{-1 + \sqrt{65}}{8} \)
2. \( \cos \theta = \frac{-1 - \sqrt{65}}{8} \)
The second value \( \frac{-1 - \sqrt{65}}{8} \) is less than -1, which is outside the range of the cosine function. Thus, we only consider:
\[
\cos \theta = \frac{-1 + \sqrt{65}}{8}
\]
### Step 6: Find the Range of \(\theta\)
To find the number of solutions for \( \theta \) in the interval \([0, \pi]\):
- The cosine function decreases from 1 to -1 in the interval \([0, \pi]\).
- Since \( \frac{-1 + \sqrt{65}}{8} \) is a valid value for cosine (between -1 and 1), there will be exactly one solution for \( \theta \) in the interval \([0, \pi]\).
### Conclusion
The number of values of \( \theta \) in the interval \([0, \pi]\) is:
\[
\boxed{1}
\]
