To determine the value of [tex]\( y \)[/tex] for the given equation
[tex]\[
\sin(x + y) = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x),
\][/tex]
we can use the identity for the sine of a sum of two angles. Let us compare the given equation to a known angle identity.
The sum of angles formula for sine is:
[tex]\[
\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B).
\][/tex]
We want our given equation to match this form. Notice that:
[tex]\[
\sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y),
\][/tex]
The given equation is:
[tex]\[
\sin(x + y) = \frac{\sqrt{3}}{2} \sin(x) + \frac{1}{2} \cos(x).
\][/tex]
By comparing the two forms, we see:
1. [tex]\(\sin(x + y)\)[/tex] corresponds to [tex]\(\sin(x) \cos(y) + \cos(x) \sin(y)\)[/tex].
2. [tex]\(\frac{\sqrt{3}}{2} \sin(x)\)[/tex] corresponds to [tex]\(\cos(y) \sin(x)\)[/tex].
3. [tex]\(\frac{1}{2} \cos(x)\)[/tex] corresponds to [tex]\(\cos(x) \sin(y)\)[/tex].
Let us focus on these individual components. The coefficient of [tex]\(\sin(x)\)[/tex] in the given equation is [tex]\(\frac{\sqrt{3}}{2}\)[/tex], which implies:
[tex]\[
\cos(y) = \frac{\sqrt{3}}{2}.
\][/tex]
Similarly, the coefficient of [tex]\(\cos(x)\)[/tex] in the given equation is [tex]\(\frac{1}{2}\)[/tex], which implies:
[tex]\[
\sin(y) = \frac{1}{2}.
\][/tex]
We need to find [tex]\( y \)[/tex] such that both [tex]\(\cos(y) = \frac{\sqrt{3}}{2}\)[/tex] and [tex]\(\sin(y) = \frac{1}{2}\)[/tex]. These are known values where:
[tex]\[
y = \frac{\pi}{3}.
\][/tex]
Thus, the value of [tex]\( y \)[/tex] that satisfies the given equation is:
[tex]\[
\boxed{\frac{\pi}{3}}.
\][/tex]
