The number of points in interval `[ - (pi)/(2) , (pi)/(2)], ` where the graphs of the curves ` y = cos x ` and ` y= sin 3x , -(pi)/(2) le x le (pi)/(2) ` intersects is

Answer

To find the number of points in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) where the graphs of the curves \(y = \cos x\) and \(y = \sin 3x\) intersect, we need to solve the equation:

\[ \cos x = \sin 3x \]

### Step 1: Rewrite the equation
Using the identity \(\sin 3x = \cos\left(\frac{\pi}{2} - 3x\right)\), we can rewrite the equation as:

\[ \cos x = \cos\left(\frac{\pi}{2} - 3x\right) \]

### Step 2: Set up the cosine equation
From the property of cosine, we know that if \(\cos A = \cos B\), then:

\[ A = 2n\pi \pm B \]

where \(n\) is any integer. Therefore, we can set up two cases:

1. \(x = 2n\pi + \left(\frac{\pi}{2} - 3x\right)\)
2. \(x = 2n\pi - \left(\frac{\pi}{2} - 3x\right)\)

### Step 3: Solve the first case
For the first case:

\[ x = 2n\pi + \frac{\pi}{2} - 3x \]

Rearranging gives:

\[ 4x = 2n\pi + \frac{\pi}{2} \]

\[ x = \frac{2n\pi + \frac{\pi}{2}}{4} = \frac{n\pi}{2} + \frac{\pi}{8} \]

### Step 4: Solve for \(n = 0\) and \(n = -1\)
1. For \(n = 0\):

\[ x = \frac{0 + \frac{\pi}{2}}{4} = \frac{\pi}{8} \]

2. For \(n = -1\):

\[ x = \frac{-2\pi + \frac{\pi}{2}}{4} = \frac{-4\pi + \pi}{8} = \frac{-3\pi}{8} \]

### Step 5: Solve the second case
For the second case:

\[ x = 2n\pi - \left(\frac{\pi}{2} - 3x\right) \]

Rearranging gives:

\[ x = 2n\pi - \frac{\pi}{2} + 3x \]

\[ -2x = 2n\pi - \frac{\pi}{2} \]

\[ 2x = -2n\pi + \frac{\pi}{2} \]

\[ x = -n\pi + \frac{\pi}{4} \]

### Step 6: Solve for \(n = 0\) and \(n = -1\)
1. For \(n = 0\):

\[ x = 0 + \frac{\pi}{4} = \frac{\pi}{4} \]

2. For \(n = -1\):

\[ x = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \quad (\text{not in the interval}) \]

### Step 7: Collect all solutions
The valid solutions from both cases in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\) are:

1. \(x = \frac{\pi}{8}\)
2. \(x = -\frac{3\pi}{8}\)
3. \(x = \frac{\pi}{4}\)

### Conclusion
Thus, the number of points where the curves intersect in the given interval is **3**.

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