To find the exact value of [tex]\(\sin 75^\circ\)[/tex], we can use the sum identity for sine. We'll express [tex]\(75^\circ\)[/tex] as the sum of two angles [tex]\(45^\circ\)[/tex] and [tex]\(30^\circ\)[/tex], for which the sine and cosine values are well-known.
Using the sum identity for sine:
[tex]\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\][/tex]
We need the exact value of [tex]\(\sin (45^\circ + 30^\circ)\)[/tex]. So let's denote:
[tex]\[
A = 45^\circ \quad \text{and} \quad B = 30^\circ
\][/tex]
Plugging these into the sum identity, we get:
[tex]\[
\sin 75^\circ = \sin (45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ
\][/tex]
Now, we use the known exact trigonometric values:
[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]
Substituting these values in, we get:
[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]
Now, multiply the terms:
[tex]\[
\sin 75^\circ = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
Simplify the expression:
[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]
Thus, the exact value of [tex]\(\sin 75^\circ\)[/tex] in terms of a sum or difference identity is:
[tex]\[
\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
