To determine the exact value of [tex]\(\sin(75^\circ)\)[/tex], we can use the angle addition formula for sine:
[tex]\[
\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
\][/tex]
For [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 30^\circ\)[/tex]:
[tex]\[
\sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)
\][/tex]
We know the exact values of sine and cosine for 45° and 30°:
[tex]\[
\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
\cos(30^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
Substituting these values into the angle addition formula:
[tex]\[
\sin(75^\circ) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)
\][/tex]
Simplifying this expression:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2} \cdot 1}{4} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
Now, let's match this with the given choices:
[tex]\[
\frac{1}{2},\; \frac{\sqrt{3}}{2},\; \frac{\sqrt{2 - \sqrt{3}}}{2},\; \frac{\sqrt{2 + \sqrt{3}}}{2}
\][/tex]
We have calculated [tex]\(\sin(75^\circ)\)[/tex] to be [tex]\(\frac{\sqrt{6} + \sqrt{2}}{4}\)[/tex]. To find the exact match among the choices, we test and simplify each choice by substituting back the possible exact values.
Upon close inspection and evaluation of these results, it turns out:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}
\][/tex]
Thus, the exact value of [tex]\(\sin(75^\circ)\)[/tex] is:
[tex]\[
\boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}}
\][/tex]
