To express the difference of sines [tex]\(\sin(15.4y) - \sin(11.6y)\)[/tex] as a product, we use the sum-to-product identities.
The specific identity we use is:
[tex]\[
\sin(A) - \sin(B) = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)
\][/tex]
Substitute [tex]\(A = 15.4y\)[/tex] and [tex]\(B = 11.6y\)[/tex]:
1. Calculate the average of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
\frac{A + B}{2} = \frac{15.4y + 11.6y}{2} = \frac{27y}{2} = 13.5y
\][/tex]
2. Calculate the half-difference of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
\frac{A - B}{2} = \frac{15.4y - 11.6y}{2} = \frac{3.8y}{2} = 1.9y
\][/tex]
Now, substitute these values into the identity:
[tex]\[
\sin(15.4y) - \sin(11.6y) = 2 \cos\left(\frac{15.4y + 11.6y}{2}\right) \sin\left(\frac{15.4y - 11.6y}{2}\right)
\][/tex]
[tex]\[
= 2 \cos(13.5y) \sin(1.9y)
\][/tex]
Therefore, the expression [tex]\(\sin(15.4y) - \sin(11.6y)\)[/tex] as a product is:
[tex]\[
2 \cos(13.5y) \sin(1.9y)
\][/tex]
