To transform the sum of two sine functions into a product, we can use the sum-to-product identities. Specifically, the identity we need is:
[tex]\[
\sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)
\][/tex]
Given the expression:
[tex]\[
\sin(22.4p) + \sin(10.4p)
\][/tex]
Let's identify [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
A = 22.4p \quad \text{and} \quad B = 10.4p
\][/tex]
We can now apply the sum-to-product formula. First, we compute [tex]\(\frac{A + B}{2}\)[/tex] and [tex]\(\frac{A - B}{2}\)[/tex].
1. Calculate [tex]\(\frac{A + B}{2}\)[/tex]:
[tex]\[
\frac{22.4p + 10.4p}{2} = \frac{32.8p}{2} = 16.4p
\][/tex]
2. Calculate [tex]\(\frac{A - B}{2}\)[/tex]:
[tex]\[
\frac{22.4p - 10.4p}{2} = \frac{12p}{2} = 6.0p
\][/tex]
Using the sum-to-product identity, substitute these values into the formula:
[tex]\[
\sin(22.4p) + \sin(10.4p) = 2 \sin(16.4p) \cos(6.0p)
\][/tex]
Therefore, the given sum of sines can be written as the product:
[tex]\[
\sin(22.4p) + \sin(10.4p) = 2 \sin(16.4p) \cos(6.0p)
\][/tex]
