Simplify by reducing to a single term :
`sin 22 ^(@) cos 38^(@) + cos 22^(@) sin 38^(@).`

Answer

To simplify the expression \( \sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ \), we can use the sine addition formula. Here are the steps to solve the problem:

### Step 1: Identify the terms
The given expression is:
\[ \sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ \]

### Step 2: Recognize the sine addition formula
Recall the sine addition formula:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
In this case, we can let \( a = 22^\circ \) and \( b = 38^\circ \).

### Step 3: Apply the sine addition formula
Using the sine addition formula, we can rewrite the expression:
\[ \sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ = \sin(22^\circ + 38^\circ) \]

### Step 4: Calculate the angle
Now, calculate \( 22^\circ + 38^\circ \):
\[ 22^\circ + 38^\circ = 60^\circ \]

### Step 5: Write the simplified expression
Thus, we have:
\[ \sin(22^\circ + 38^\circ) = \sin 60^\circ \]

### Step 6: Find the value of \( \sin 60^\circ \)
We know that:
\[ \sin 60^\circ = \frac{\sqrt{3}}{2} \]

### Final Result
Therefore, the simplified expression is:
\[ \sin 22^\circ \cos 38^\circ + \cos 22^\circ \sin 38^\circ = \frac{\sqrt{3}}{2} \]

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