To simplify the expression [tex]\(\sin 38^\circ + \sin 22^\circ\)[/tex] using the sum-to-product identity, we follow these steps:
1. Sum-to-Product Formula for Sine:
The sum-to-product identity for sine states:
[tex]\[
\sin x + \sin y = 2 \sin \left( \frac{x + y}{2} \right) \cos \left( \frac{x - y}{2} \right)
\][/tex]
2. Assign the Values:
Let [tex]\(x = 38^\circ\)[/tex] and [tex]\(y = 22^\circ\)[/tex]. We want to simplify [tex]\(\sin 38^\circ + \sin 22^\circ\)[/tex].
3. Calculate the Intermediate Angles:
First, we need to find:
[tex]\[
\frac{x + y}{2} = \frac{38^\circ + 22^\circ}{2} = \frac{60^\circ}{2} = 30^\circ
\][/tex]
And,
[tex]\[
\frac{x - y}{2} = \frac{38^\circ - 22^\circ}{2} = \frac{16^\circ}{2} = 8^\circ
\][/tex]
4. Rewrite the Expression Using the Identity:
Substituting these values into the sum-to-product formula, we get:
[tex]\[
\sin 38^\circ + \sin 22^\circ = 2 \sin \left(30^\circ\right) \cos \left(8^\circ\right)
\][/tex]
5. Interpret the Expression:
Notice that in the problem statement, we are given that [tex]\(\sin 38^\circ + \sin 22^\circ = \sin A^\circ\)[/tex]. Thus, we need to determine the angle [tex]\(A\)[/tex] such that:
[tex]\[
\sin A^\circ = 2 \sin \left(30^\circ\right) \cos \left(8^\circ\right)
\][/tex]
6. Simplify Further Using Trigonometric Values:
We know that:
[tex]\[
\sin 30^\circ = \frac{1}{2}
\][/tex]
Therefore:
[tex]\[
2 \sin \left(30^\circ\right) \cos \left(8^\circ\right) = 2 \left(\frac{1}{2}\right) \cos \left(8^\circ\right) = \cos \left(8^\circ\right)
\][/tex]
Since we have simplified to an angle equal relationship, we conclude:
[tex]\[
\sin 38^\circ + \sin 22^\circ = \sin 30^\circ
\][/tex]
7. Final Answer:
Therefore, the angle [tex]\(A\)[/tex] is:
[tex]\[
A = 30^\circ
\][/tex]
So, [tex]\(A = 30^\circ\)[/tex].
