Let `p,qin N and q gt p`, the number of solutions of the equation `q|sin theta |=p|cos theta| ` in the interval `[0,2pi]` is

Answer

To solve the equation \( q |\sin \theta| = p |\cos \theta| \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ q |\sin \theta| = p |\cos \theta| \]
Dividing both sides by \( |\cos \theta| \) (assuming \( \cos \theta \neq 0 \)), we get:
\[ \frac{|\sin \theta|}{|\cos \theta|} = \frac{p}{q} \]
This simplifies to:
\[ |\tan \theta| = \frac{p}{q} \]

### Step 2: Analyze the value of \( \frac{p}{q} \)
Since \( p \) and \( q \) are natural numbers and \( q > p \), we know that:
\[ 0 < \frac{p}{q} < 1 \]

### Step 3: Graph the function
Next, we analyze the function \( |\tan \theta| \) over the interval \( [0, 2\pi] \). The function \( \tan \theta \) has vertical asymptotes at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \). The graph of \( |\tan \theta| \) will look like this:

- From \( 0 \) to \( \frac{\pi}{2} \), \( |\tan \theta| \) increases from \( 0 \) to \( +\infty \).
- From \( \frac{\pi}{2} \) to \( \frac{3\pi}{2} \), \( |\tan \theta| \) decreases from \( +\infty \) to \( 0 \).
- From \( \frac{3\pi}{2} \) to \( 2\pi \), \( |\tan \theta| \) increases again from \( 0 \) to \( +\infty \).

### Step 4: Find intersections
Now we need to find the intersections of the line \( y = \frac{p}{q} \) with the graph of \( |\tan \theta| \).

Since \( 0 < \frac{p}{q} < 1 \), the line \( y = \frac{p}{q} \) will intersect the graph of \( |\tan \theta| \) in each of the intervals:
- \( [0, \frac{\pi}{2}) \)
- \( (\frac{\pi}{2}, \frac{3\pi}{2}) \)
- \( [\frac{3\pi}{2}, 2\pi) \)

### Step 5: Count the solutions
In each of these intervals, the line \( y = \frac{p}{q} \) intersects the graph of \( |\tan \theta| \) at two points. Therefore, we have:
- 2 solutions in \( [0, \frac{\pi}{2}) \)
- 2 solutions in \( (\frac{3\pi}{2}, 2\pi) \)

Thus, the total number of solutions in the interval \( [0, 2\pi] \) is:
\[ 2 + 2 = 4 \]

### Final Answer
The number of solutions of the equation \( q |\sin \theta| = p |\cos \theta| \) in the interval \( [0, 2\pi] \) is \( \boxed{4} \).