To determine the values of \(\theta\) for which the points \(P(\sin\theta, 0)\) and \(Q(0, \cos\theta)\) lie on the same side of the line given by the equation \(2x - 3y = 0\), we can follow these steps:
### Step 1: Understand the Line Equation
The line equation can be rewritten in slope-intercept form:
\[
3y = 2x \implies y = \frac{2}{3}x
\]
This line passes through the origin (0,0) and has a slope of \(\frac{2}{3}\).
### Step 2: Identify the Points
The points are defined as:
- \(P(\sin\theta, 0)\) lies on the x-axis.
- \(Q(0, \cos\theta)\) lies on the y-axis.
### Step 3: Determine the Side of the Line
To find out on which side of the line the points lie, we can substitute the coordinates of points \(P\) and \(Q\) into the line equation \(2x - 3y = 0\).
1. For point \(P(\sin\theta, 0)\):
\[
2(\sin\theta) - 3(0) = 2\sin\theta
\]
- If \(2\sin\theta > 0\), then \(P\) is above the line.
- If \(2\sin\theta < 0\), then \(P\) is below the line.
2. For point \(Q(0, \cos\theta)\):
\[
2(0) - 3(\cos\theta) = -3\cos\theta
\]
- If \(-3\cos\theta > 0\) (or \(\cos\theta < 0\)), then \(Q\) is above the line.
- If \(-3\cos\theta < 0\) (or \(\cos\theta > 0\)), then \(Q\) is below the line.
### Step 4: Analyze the Conditions
For \(P\) and \(Q\) to lie on the same side of the line, both conditions must be satisfied:
1. **Case 1**: \(P\) is above the line and \(Q\) is above the line:
- \(2\sin\theta > 0 \implies \sin\theta > 0\) (This occurs in the first and second quadrants)
- \(-3\cos\theta > 0 \implies \cos\theta < 0\) (This occurs in the second quadrant)
Thus, in this case, \(\theta\) must be in the second quadrant.
2. **Case 2**: \(P\) is below the line and \(Q\) is below the line:
- \(2\sin\theta < 0 \implies \sin\theta < 0\) (This occurs in the third and fourth quadrants)
- \(-3\cos\theta < 0 \implies \cos\theta > 0\) (This occurs in the fourth quadrant)
Thus, in this case, \(\theta\) must be in the fourth quadrant.
### Step 5: Conclusion
From the analysis, we find that the points \(P\) and \(Q\) lie on the same side of the line if \(\theta\) is in either the second quadrant or the fourth quadrant. However, since the question specifies options and only the second quadrant is provided, the final answer is:
\[
\text{The points } P \text{ and } Q \text{ lie on the same side of the line if } \theta \text{ lies in the second quadrant.}
\]
