Convert the sums or differences into products:
`sin 37 ^(@) + sin 21 ^(@)`

Answer

To convert the sum of sines into a product, we can use the sine addition formula. Here’s the step-by-step solution:

### Step 1: Identify the angles
We have the expression:
\[ \sin 37^\circ + \sin 21^\circ \]

### Step 2: Use the sine addition formula
The formula for the sum of two sine functions is:
\[ \sin C + \sin D = 2 \sin\left(\frac{C + D}{2}\right) \cos\left(\frac{C - D}{2}\right) \]
In our case, \(C = 37^\circ\) and \(D = 21^\circ\).

### Step 3: Calculate \(C + D\) and \(C - D\)
Now, we calculate:
\[ C + D = 37^\circ + 21^\circ = 58^\circ \]
\[ C - D = 37^\circ - 21^\circ = 16^\circ \]

### Step 4: Substitute into the formula
Now we substitute these values into the sine addition formula:
\[ \sin 37^\circ + \sin 21^\circ = 2 \sin\left(\frac{58^\circ}{2}\right) \cos\left(\frac{16^\circ}{2}\right) \]

### Step 5: Simplify the expression
Calculating the halves:
\[ \frac{58^\circ}{2} = 29^\circ \]
\[ \frac{16^\circ}{2} = 8^\circ \]
So we can rewrite the expression as:
\[ \sin 37^\circ + \sin 21^\circ = 2 \sin(29^\circ) \cos(8^\circ) \]

### Final Answer
Thus, the sum of \(\sin 37^\circ + \sin 21^\circ\) can be expressed as:
\[ \sin 37^\circ + \sin 21^\circ = 2 \sin(29^\circ) \cos(8^\circ) \]

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