Find the coodinates of the point of intersection of the curves `y= cos x , y= sin 3x ` if `-(pi)/(2) le x le (pi)/(2)`

Answer

To find the coordinates of the points of intersection of the curves \( y = \cos x \) and \( y = \sin 3x \) within the interval \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \), we will follow these steps:

### Step 1: Set the equations equal to each other
To find the points of intersection, we set the two equations equal:
\[ \cos x = \sin 3x \]

### Step 2: Use the complementary angle identity
Using the identity \( \cos x = \sin\left(\frac{\pi}{2} - x\right) \), we can rewrite the equation:
\[ \sin\left(\frac{\pi}{2} - x\right) = \sin 3x \]

### Step 3: Apply the sine difference identity
Using the sine difference identity \( \sin A - \sin B = 0 \), we can express this as:
\[ \sin 3x - \sin\left(\frac{\pi}{2} - x\right) = 0 \]
This can be rewritten as:
\[ \sin 3x - \sin\left(\frac{\pi}{2} - x\right) = 0 \]

### Step 4: Use the sine subtraction formula
Using the formula \( \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \), we have:
\[ 2 \cos\left(\frac{3x + \left(\frac{\pi}{2} - x\right)}{2}\right) \sin\left(\frac{3x - \left(\frac{\pi}{2} - x\right)}{2}\right) = 0 \]

### Step 5: Simplify the expressions
Calculating the averages:
\[ \frac{3x + \frac{\pi}{2} - x}{2} = \frac{2x + \frac{\pi}{2}}{2} = x + \frac{\pi}{4} \]
\[ \frac{3x - \left(\frac{\pi}{2} - x\right)}{2} = \frac{4x - \frac{\pi}{2}}{2} = 2x - \frac{\pi}{4} \]
Thus, we have:
\[ 2 \cos\left(x + \frac{\pi}{4}\right) \sin\left(2x - \frac{\pi}{4}\right) = 0 \]

### Step 6: Set each factor to zero
Setting each factor to zero gives us two equations:
1. \( \cos\left(x + \frac{\pi}{4}\right) = 0 \)
2. \( \sin\left(2x - \frac{\pi}{4}\right) = 0 \)

### Step 7: Solve the first equation
For \( \cos\left(x + \frac{\pi}{4}\right) = 0 \):
\[ x + \frac{\pi}{4} = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Solving for \( x \):
\[ x = \frac{\pi}{2} - \frac{\pi}{4} + n\pi = \frac{\pi}{4} + n\pi \]
Considering \( n = 0 \):
\[ x = \frac{\pi}{4} \]

### Step 8: Solve the second equation
For \( \sin\left(2x - \frac{\pi}{4}\right) = 0 \):
\[ 2x - \frac{\pi}{4} = n\pi \quad (n \in \mathbb{Z}) \]
Solving for \( x \):
\[ 2x = n\pi + \frac{\pi}{4} \implies x = \frac{n\pi}{2} + \frac{\pi}{8} \]
Considering \( n = 0 \):
\[ x = \frac{\pi}{8} \]
Considering \( n = -1 \):
\[ x = -\frac{\pi}{2} + \frac{\pi}{8} = -\frac{4\pi}{8} + \frac{\pi}{8} = -\frac{3\pi}{8} \]

### Step 9: Find corresponding \( y \) values
Now we find the \( y \) coordinates for the \( x \) values found:
1. For \( x = \frac{\pi}{4} \):
\[ y = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
Point: \( \left(\frac{\pi}{4}, \frac{1}{\sqrt{2}}\right) \)

2. For \( x = \frac{\pi}{8} \):
\[ y = \cos\left(\frac{\pi}{8}\right) \]
Point: \( \left(\frac{\pi}{8}, \cos\left(\frac{\pi}{8}\right)\right) \)

3. For \( x = -\frac{3\pi}{8} \):
\[ y = \cos\left(-\frac{3\pi}{8}\right) = \cos\left(\frac{3\pi}{8}\right) \]
Point: \( \left(-\frac{3\pi}{8}, \cos\left(\frac{3\pi}{8}\right)\right) \)

### Final Points of Intersection
Thus, the points of intersection are:
1. \( \left(\frac{\pi}{4}, \frac{1}{\sqrt{2}}\right) \)
2. \( \left(\frac{\pi}{8}, \cos\left(\frac{\pi}{8}\right)\right) \)
3. \( \left(-\frac{3\pi}{8}, \cos\left(\frac{3\pi}{8}\right)\right) \)