To find the exact value of [tex]\(\sin\left(105^\circ\right)\)[/tex], we can use the angle addition formula for sine. The angle [tex]\(105^\circ\)[/tex] can be written as [tex]\(105^\circ = 45^\circ + 60^\circ\)[/tex].
The sine addition formula is given by:
[tex]\[
\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
\][/tex]
For [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 60^\circ\)[/tex]:
1. Find [tex]\(\sin(45^\circ)\)[/tex] and [tex]\(\cos(45^\circ)\)[/tex]:
[tex]\[
\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
2. Find [tex]\(\sin(60^\circ)\)[/tex] and [tex]\(\cos(60^\circ)\)[/tex]:
[tex]\[
\sin(60^\circ) = \frac{\sqrt{3}}{2} \quad \text{and} \quad \cos(60^\circ) = \frac{1}{2}
\][/tex]
Substitute these values into the sine addition formula:
[tex]\[
\sin(105^\circ) = \sin(45^\circ + 60^\circ) = \sin(45^\circ)\cos(60^\circ) + \cos(45^\circ)\sin(60^\circ)
\][/tex]
Now calculate each part:
[tex]\[
\sin(45^\circ)\cos(60^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{2}}{4}
\][/tex]
[tex]\[
\cos(45^\circ)\sin(60^\circ) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{2}\sqrt{3}}{4} = \frac{\sqrt{6}}{4}
\][/tex]
Adding these together:
[tex]\[
\sin(105^\circ) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}
\][/tex]
Simplify the fraction:
[tex]\[
\sin(105^\circ) = \frac{\sqrt{2 + \sqrt{3} + 2 - \sqrt{3}}}{2} = \frac{\sqrt{(2 + \sqrt{3})(2 - \sqrt{3})}}{2}
\][/tex]
Given the options, we can see that the correct expression for [tex]\(\sin\left(105^\circ\right)\)[/tex] matches:
[tex]\[
\sin(105^\circ) = \frac{\sqrt{2 + \sqrt{3}}}{2}
\][/tex]
Thus, the exact value of [tex]\(\sin(105^\circ)\)[/tex] is:
[tex]\[
\frac{\sqrt{2 + \sqrt{3}}}{2}
\][/tex]
