To find the exact value of [tex]\(\sin 75^\circ\)[/tex], we can use the angle addition formula instead of a half-angle identity. Let's break it down step-by-step:
1. Represent 75 degrees as a sum of known angles:
[tex]\[
75^\circ = 45^\circ + 30^\circ
\][/tex]
2. Use the angle addition formula for sine:
[tex]\[
\sin(75^\circ) = \sin(45^\circ + 30^\circ)
\][/tex]
The angle addition formula for sine is:
[tex]\[
\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)
\][/tex]
Let [tex]\(a = 45^\circ\)[/tex] and [tex]\(b = 30^\circ\)[/tex]:
[tex]\[
\sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ)
\][/tex]
3. Plug in the known values of sine and cosine for the angles:
- [tex]\(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)[/tex]
- [tex]\(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)[/tex]
- [tex]\(\sin(30^\circ) = \frac{1}{2}\)[/tex]
So:
[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]
4. Simplify the expression:
Multiply the corresponding terms:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
\][/tex]
Combine the terms under a common denominator:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}
\][/tex]
5. Match the simplified form to the given choices:
We need to simplify further to recognize the correct option:
- The numerator [tex]\(\sqrt{2+\sqrt{3}}\)[/tex] is equivalent to [tex]\(\sqrt{6} + \sqrt{2}\)[/tex] when considering rationalization and basic trigonometric identities.
By simplifying the exact value form, we get:
[tex]\[
\sin(75^\circ) = \frac{\sqrt{2+\sqrt{3}}}{2}
\][/tex]
Comparing this result, we see that it matches the given option exactly:
Given the choices:
a. [tex]\(\frac{\sqrt{2+\sqrt{3}}}{2}\)[/tex]
c. [tex]\(\sqrt{2+\sqrt{3}}\)[/tex]
b. [tex]\(\frac{\sqrt{2-\sqrt{3}}}{2}\)[/tex]
d. [tex]\(\sqrt{2-\sqrt{3}}\)[/tex]
The correct answer is option A:
[tex]\[
\boxed{\frac{\sqrt{2+\sqrt{3}}}{2}}
\][/tex]
