Evaluate the expression.
$\cos^{-1}\left(-\frac{\sqrt2}{2}\right)$

• A. $\frac{\pi }{4}$
• B. $\frac{3\pi }{4}$
• C. $\frac{5\pi }{4}$
• D. $\frac{7\pi }{4}$

Answer: (B)

Verified step by step guidance

1. Understand that the problem involves finding the angle whose cosine is \(-\frac{\sqrt{2}}{2}\). This is the inverse cosine function, \(\cos^{-1}\).
2. Recall that the cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
3. Recognize that \(-\frac{\sqrt{2}}{2}\) is a known cosine value corresponding to angles in the second and third quadrants, specifically \(\frac{3\pi}{4}\) and \(\frac{5\pi}{4}\).
4. Since \(\cos^{-1}\) returns values in the range \([0, \pi]\), the angle \(\frac{3\pi}{4}\) is the correct answer within this range.
5. Verify that \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\) to ensure the solution is correct.