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]
We choose [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 30^\circ\)[/tex].
First, recall the sine and cosine values for the angles [tex]\(45^\circ\)[/tex] and [tex]\(30^\circ\)[/tex]:
- [tex]\(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin(30^\circ) = \frac{1}{2}\)[/tex]
- [tex]\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)[/tex]
Using the angle addition formula:
[tex]\[ \sin(75^\circ) = \sin(45^\circ + 30^\circ) \][/tex]
[tex]\[ \sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) \][/tex]
Substitute in the known sine and cosine values:
[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]
[tex]\[ \sin(75^\circ) = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2} \cdot 1}{4} \][/tex]
[tex]\[ \sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \][/tex]
[tex]\[ \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
We need to determine the exact simplified form of this expression.
The exact value of [tex]\(\sin(75^\circ)\)[/tex] can be recognized as one of the given options based on standard trigonometric identities, and after comparing numerically, we find it matches:
[tex]\[ \sin(75^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2} \][/tex]
Therefore, the correct exact value is:
[tex]\[ \sin(75^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2} \][/tex]
Hence, the exact value of [tex]\(\sin(75^\circ)\)[/tex] is [tex]\(\boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}}\)[/tex].
