Find the exact value using a sum or difference identity.

[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]

[tex]\[
\sin 75^{\circ} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}
\][/tex]

[tex]\[
\sin 75^{\circ} = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]

Hint:
[tex]\[
\sin (A \pm B) = \sin A \cos B \pm \cos A \sin B
\][/tex]
[tex]\[
\cos (A \pm B) = \cos A \cos B \mp \sin A \sin B
\][/tex]



Answer :

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]