For `-(pi)/(2)le x le (pi)/(2)`, the number of point of intersection of curves `y= cos x and y = sin 3x` is

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of points of intersection of the curves \( y = \cos x \) and \( y = \sin 3x \) for \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \), we need to solve the equation:

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

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

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

### Step 2: Set up the general solutions
The sine function has the property that if \( \sin A = \sin B \), then:

\[ A = B + 2n\pi \quad \text{or} \quad A = \pi - B + 2n\pi \]

Applying this to our equation, we have two cases:

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

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

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

Rearranging gives:

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

### Step 4: Solve the second case
From the second case:

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

Rearranging gives:

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

### Step 5: Find valid solutions in the interval
Now we need to find integer values of \( n \) such that \( x \) falls within the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).

**For the first case:**

- \( x = \frac{\pi}{8} - \frac{n\pi}{2} \)

1. For \( n = 0 \): \( x = \frac{\pi}{8} \) (valid)
2. For \( n = -1 \): \( x = \frac{\pi}{8} + \frac{\pi}{2} = \frac{5\pi}{8} \) (not valid)
3. For \( n = 1 \): \( x = \frac{\pi}{8} - \frac{\pi}{2} = -\frac{3\pi}{8} \) (valid)

**For the second case:**

- \( x = n\pi + \frac{\pi}{4} \)

1. For \( n = 0 \): \( x = \frac{\pi}{4} \) (valid)
2. For \( n = -1 \): \( x = -\frac{\pi}{4} \) (valid)
3. For \( n = 1 \): \( x = \frac{5\pi}{4} \) (not valid)

### Step 6: List valid solutions
The valid solutions are:

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

### Conclusion
Thus, the total number of points of intersection of the curves \( y = \cos x \) and \( y = \sin 3x \) in the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \) is **3**.

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