Solve ` sin 5 x = cos 2 x `.

Answer

To solve the equation \( \sin 5x = \cos 2x \), we can use the identity that relates sine and cosine.

### Step 1: Use the identity
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
In radians, this can be expressed as:
\[ \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) \]
Thus, we can rewrite the equation as:
\[ \sin 5x = \sin\left(\frac{\pi}{2} - 2x\right) \]

### Step 2: Set the arguments equal
Since the sine function is periodic, we can set the arguments equal to each other:
\[ 5x = \frac{\pi}{2} - 2x \]

### Step 3: Solve for \( x \)
Now, we can rearrange the equation to isolate \( x \):
\[ 5x + 2x = \frac{\pi}{2} \]
\[ 7x = \frac{\pi}{2} \]
Now, divide both sides by 7:
\[ x = \frac{\pi}{14} \]

### Final Answer
Thus, the solution to the equation \( \sin 5x = \cos 2x \) is:
\[ x = \frac{\pi}{14} \]