To find the exact value of [tex]\(\sin \frac{3 \pi}{4}\)[/tex], we can use the following trigonometric identity and properties of the sine function.
1. Recognize the angle in terms of [tex]\(\pi\)[/tex]: [tex]\(\frac{3 \pi}{4}\)[/tex] is an angle in radians.
2. Use the sine angle subtraction identity:
[tex]\[
\sin \left(\frac{3 \pi}{4}\right) = \sin \left(\pi - \frac{\pi}{4}\right)
\][/tex]
3. Apply the sine identity: There is a trigonometric identity that states:
[tex]\[
\sin(\pi - x) = \sin(x)
\][/tex]
Using this identity, we can simplify the sine of the given angle:
[tex]\[
\sin \left(\pi - \frac{\pi}{4}\right) = \sin \left(\frac{\pi}{4}\right)
\][/tex]
4. Evaluate [tex]\(\sin \frac{\pi}{4}\)[/tex]: The exact value of [tex]\(\sin \frac{\pi}{4}\)[/tex] is well known from trigonometric tables or unit circle values:
[tex]\[
\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}
\][/tex]
Therefore, combining all steps, we have:
[tex]\[
\sin \frac{3 \pi}{4} = \frac{\sqrt{2}}{2}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{2}}{2}}
\][/tex]
