To solve for [tex]\(\sin \left(\frac{3 \pi}{4}\right)\)[/tex], we'll need to determine the sine of the given angle, [tex]\(\frac{3 \pi}{4}\)[/tex].
Firstly, it helps to understand where [tex]\(\frac{3\pi}{4}\)[/tex] radians lies on the unit circle. This angle is in the second quadrant, as it is between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\pi\)[/tex]. The reference angle in this case is [tex]\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)[/tex].
In the second quadrant, the sine function is positive. Therefore, the value of [tex]\(\sin\left( \frac{3\pi}{4} \right) \)[/tex] will be the same as the sine of the reference angle [tex]\(\frac{\pi}{4}\)[/tex], but positive since sine is positive in the second quadrant.
We know that the sine of [tex]\(\frac{\pi}{4}\)[/tex] is [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
Therefore, [tex]\(\sin \left(\frac{3 \pi}{4}\right)\)[/tex] is also [tex]\(\frac{\sqrt{2}}{2}\)[/tex].
Comparing this with the given options:
A. [tex]\(\frac{1}{2}\)[/tex]
B. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
C. [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]
D. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
We see that [tex]\(\sin \left(\frac{3 \pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{2}}{2}}
\][/tex]
