To find the exact value of \(\cos \frac{3\pi}{4}\), we can follow these steps:
1. Recognize the Angle in the Unit Circle:
The angle \(\frac{3\pi}{4}\) is in the second quadrant of the unit circle.
2. Reference Angle:
The reference angle for \(\frac{3\pi}{4}\) is \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\). This is useful because the cosine and sine functions have known values at \(\frac{\pi}{4}\).
3. Cosine Function in the Second Quadrant:
Cosine is negative in the second quadrant. Therefore,
[tex]\[
\cos \frac{3\pi}{4} = - \cos \frac{\pi}{4}
\][/tex]
4. Value at \(\frac{\pi}{4}\):
We know from trigonometric identities that
[tex]\[
\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}
\][/tex]
5. Apply the Sign:
Since we determined that the cosine function is negative in the second quadrant:
[tex]\[
\cos \frac{3\pi}{4} = - \cos \frac{\pi}{4} = - \frac{\sqrt{2}}{2}
\][/tex]
Thus,
[tex]\[
\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}
\][/tex]
6. Verification with Numerical Result:
The numerical value of \(-\frac{\sqrt{2}}{2}\) is approximately \(-0.7071067811865476\), which confirms our result.
Therefore, the exact value of \(\cos \frac{3\pi}{4}\) is:
[tex]\[
\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}
\][/tex]
