To find the exact value of [tex]\(\sin(75^\circ)\)[/tex], we can use the angle addition formula for sine. We know that [tex]\(75^\circ\)[/tex] can be written as [tex]\(45^\circ + 30^\circ\)[/tex]. Therefore, we can use the formula for the sine of the sum of two angles:
[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], this gives:
[tex]\[
\sin(75^\circ) = \sin(45^\circ) \cos(30^\circ) + \cos(45^\circ) \sin(30^\circ)
\][/tex]
Now, we need the exact values for the sine and cosine of [tex]\(45^\circ\)[/tex] and [tex]\(30^\circ\)[/tex]:
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \sin(30^\circ) = \frac{1}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
Now substitute these values into our formula:
[tex]\[
\sin(75^\circ) = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)
\][/tex]
Simplify each term:
[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]
Combine the fractions:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
By considering the given answers and the form of our result:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} = \frac{\sqrt{2+\sqrt{3}}}{2}
\][/tex]
So, the exact value of [tex]\(\sin(75^\circ)\)[/tex] is
[tex]\[
\frac{\sqrt{2+\sqrt{3}}}{2}
\][/tex]
