Let `0

Options

A. `6pi`
B. `7pi`
C. `8pi`
D. `4pi`

Explanation

To solve the equation \(3 + 3 \cos \theta = 2 \sin^2 \theta\) and find the value of \(\theta_3 + \theta_7\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 3 + 3 \cos \theta = 2 \sin^2 \theta \]
Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), we can rewrite the equation as:
\[ 3 + 3 \cos \theta = 2(1 - \cos^2 \theta) \]

### Step 2: Simplify the equation
Expanding the right side gives:
\[ 3 + 3 \cos \theta = 2 - 2 \cos^2 \theta \]
Rearranging the equation, we get:
\[ 2 \cos^2 \theta + 3 \cos \theta + 1 = 0 \]

### Step 3: Solve the quadratic equation
This is a standard quadratic equation in \(\cos \theta\). We can use the quadratic formula:
\[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 2\), \(b = 3\), and \(c = 1\). Plugging in these values:
\[ \cos \theta = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]
\[ = \frac{-3 \pm \sqrt{9 - 8}}{4} \]
\[ = \frac{-3 \pm 1}{4} \]

### Step 4: Calculate the roots
Calculating the two possible values:
1. \(\cos \theta = \frac{-2}{4} = -\frac{1}{2}\)
2. \(\cos \theta = \frac{-4}{4} = -1\)

### Step 5: Find the angles corresponding to the cosine values
1. For \(\cos \theta = -\frac{1}{2}\):
- The angles are \(\theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3}, \ldots\)

2. For \(\cos \theta = -1\):
- The angle is \(\theta = \pi, 3\pi, 5\pi, 7\pi, \ldots\)

### Step 6: List the positive solutions in increasing order
Combining the solutions:
- From \(\cos \theta = -\frac{1}{2}\): \(\frac{2\pi}{3}, \frac{4\pi}{3}, \frac{8\pi}{3}, \frac{10\pi}{3}, \ldots\)
- From \(\cos \theta = -1\): \(\pi, 3\pi, 5\pi, 7\pi, \ldots\)

Arranging them in increasing order:
1. \(\frac{2\pi}{3}\)
2. \(\pi\)
3. \(\frac{4\pi}{3}\)
4. \(3\pi\)
5. \(\frac{8\pi}{3}\)
6. \(5\pi\)
7. \(\frac{10\pi}{3}\)
8. \(7\pi\)
9. \(\frac{14\pi}{3}\)

### Step 7: Identify \(\theta_3\) and \(\theta_7\)
From the ordered list:
- \(\theta_3 = \frac{4\pi}{3}\)
- \(\theta_7 = \frac{10\pi}{3}\)

### Step 8: Calculate \(\theta_3 + \theta_7\)
Now we can add these two angles:
\[ \theta_3 + \theta_7 = \frac{4\pi}{3} + \frac{10\pi}{3} = \frac{14\pi}{3} \]

### Step 9: Final answer
The value of \(\theta_3 + \theta_7\) is:
\[ \frac{14\pi}{3} = 6\pi \]

### Conclusion
Thus, the answer is \(6\pi\).

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