Sure, let's find the exact value of [tex]\(\cos \frac{3\pi}{4}\)[/tex].
1. First, let's understand the position of the angle [tex]\(\frac{3\pi}{4}\)[/tex] in the unit circle. The angle [tex]\(\frac{3\pi}{4}\)[/tex] is in the second quadrant because it is between [tex]\(\pi/2\)[/tex] and [tex]\(\pi\)[/tex].
2. Recall that in the second quadrant, the cosine of an angle is negative. This is because the x-coordinates of the points on the unit circle in this quadrant are negative.
3. The reference angle for [tex]\(\frac{3\pi}{4}\)[/tex] is [tex]\(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\)[/tex]. The reference angle helps us determine the cosine value based on known values of standard angles.
4. The cosine of [tex]\(\frac{\pi}{4}\)[/tex] is a known value. [tex]\(\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)[/tex].
5. Since we are in the second quadrant, where cosine values are negative, we take the negative of the reference angle's cosine value.
6. Therefore, [tex]\(\cos \frac{3\pi}{4} = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2}\)[/tex].
Hence, the exact value of [tex]\(\cos \frac{3\pi}{4}\)[/tex] is:
[tex]\[
\boxed{-\frac{\sqrt{2}}{2}}
\][/tex]
