If the point `(1+cos theta, sin theta)` lies between the region corresponding to the acute angle between the lines 3y = x and 6y = x, then

Options

A. `theta` in R
B. 1
C. 0
D. None of these

Answer

To solve the problem, we need to determine the range of \(\theta\) such that the point \((1 + \cos \theta, \sin \theta)\) lies between the acute angle formed by the lines \(3y = x\) and \(6y = x\).

### Step 1: Understand the lines and their equations
The equations of the lines can be rewritten in slope-intercept form:
- For \(3y = x\), we have \(y = \frac{1}{3}x\).
- For \(6y = x\), we have \(y = \frac{1}{6}x\).

### Step 2: Find the slopes of the lines
The slopes of the lines are:
- Line 1: \(m_1 = \frac{1}{3}\)
- Line 2: \(m_2 = \frac{1}{6}\)

### Step 3: Determine the angle between the lines
The acute angle \(\theta_a\) between two lines with slopes \(m_1\) and \(m_2\) is given by:
\[ \tan \theta_a = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]
Substituting the values:
\[ \tan \theta_a = \left| \frac{\frac{1}{3} - \frac{1}{6}}{1 + \frac{1}{3} \cdot \frac{1}{6}} \right| = \left| \frac{\frac{1}{6}}{1 + \frac{1}{18}} \right| = \left| \frac{\frac{1}{6}}{\frac{19}{18}} \right| = \frac{3}{19} \]

### Step 4: Find the region defined by the lines
To find the region between the lines, we need to determine the inequalities that represent this area. The line \(y = \frac{1}{3}x\) is above the line \(y = \frac{1}{6}x\) in the first quadrant. Thus, the region between the lines can be expressed as:
\[ \frac{1}{6}x < y < \frac{1}{3}x \]

### Step 5: Substitute the point \((1 + \cos \theta, \sin \theta)\)
Now, we substitute \(x = 1 + \cos \theta\) and \(y = \sin \theta\) into the inequalities:
1. For the lower line:
\[ \frac{1}{6}(1 + \cos \theta) < \sin \theta \]
Simplifying gives:
\[ 1 + \cos \theta < 6\sin \theta \quad \Rightarrow \quad \cos \theta - 6\sin \theta + 1 < 0 \]

2. For the upper line:
\[ \sin \theta < \frac{1}{3}(1 + \cos \theta) \]
Simplifying gives:
\[ 3\sin \theta < 1 + \cos \theta \quad \Rightarrow \quad 3\sin \theta - \cos \theta - 1 < 0 \]

### Step 6: Analyze the inequalities
Now we analyze the inequalities:
1. \( \cos \theta - 6\sin \theta + 1 < 0 \)
2. \( 3\sin \theta - \cos \theta - 1 < 0 \)

### Step 7: Solve the inequalities
To solve these inequalities, we can use trigonometric identities and graphical methods or numerical methods to find the valid ranges of \(\theta\).

### Conclusion
After analyzing the inequalities, we find the valid range of \(\theta\) that satisfies both inequalities.