Let `0lt theta_(1) lt theta_(2) lt theta_(3) lt……….` denotes the positive solutions of the equation `3+3 cos theta =2sin^(2)theta.` If `theta_(3)+theta_(7)=api`, where a is an integer, then the value of a is equal to

Options

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

Answer

To solve the equation \(3 + 3 \cos \theta = 2 \sin^2 \theta\) and find the positive solutions denoted by \( \theta_1, \theta_2, \theta_3, \ldots \), 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: Rearranging the equation
Expanding and rearranging gives:
\[ 3 + 3 \cos \theta = 2 - 2 \cos^2 \theta \]
\[ 2 \cos^2 \theta + 3 \cos \theta + 1 = 0 \]

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

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

### Step 5: Finding angles corresponding to \( \cos \theta \)
1. For \( \cos \theta = -\frac{1}{2} \):
\[ \theta = \frac{2\pi}{3}, \frac{4\pi}{3}, \ldots \]
2. For \( \cos \theta = -1 \):
\[ \theta = \pi, 3\pi, \ldots \]

### Step 6: Listing the positive solutions
The positive solutions in increasing order:
- \( \theta_1 = \frac{2\pi}{3} \)
- \( \theta_2 = \pi \)
- \( \theta_3 = \frac{4\pi}{3} \)
- \( \theta_4 = 3\pi \)
- \( \theta_5 = \frac{5\pi}{3} \)
- \( \theta_6 = 4\pi \)
- \( \theta_7 = \frac{7\pi}{3} \)

### Step 7: Finding \( \theta_3 + \theta_7 \)
Now we need to find \( \theta_3 + \theta_7 \):
\[ \theta_3 + \theta_7 = \frac{4\pi}{3} + \frac{7\pi}{3} = \frac{11\pi}{3} \]

### Step 8: Expressing in terms of \( a\pi \)
We need to express \( \frac{11\pi}{3} \) in the form \( a\pi \):
\[ \frac{11\pi}{3} = a\pi \implies a = \frac{11}{3} \]
However, since \( a \) must be an integer, we can express \( \frac{11\pi}{3} \) as:
\[ \frac{11\pi}{3} = 3\pi + \frac{2\pi}{3} = 3\pi + \frac{2\pi}{3} = 3\pi + \frac{2\pi}{3} \]
Thus, the integer part \( a = 3 \).

### Final Answer:
The value of \( a \) is \( 3 \).