Find the exact value cos^-1(-square root 2/2)

Answer 1

To find the exact value of \(\cos^{-1}\left(-\sqrt{\frac{2}{2}}\right)\), we need to determine the angle whose cosine is \(-\sqrt{\frac{2}{2}} = -1\).

Since the cosine function is the x-coordinate of a point on the unit circle, \(\cos^{-1}(-1)\) would give us the angle whose cosine is \(-1\).

Therefore, the exact value of \(\cos^{-1}\left(-\sqrt{\frac{2}{2}}\right)\) is \(\boxed{\pi}\).

Answer 2

the principal value for cos^-1(x) is [0,pi]
so, since cos(3π/4) = -1/√2,
cos^-1(-1/√2) = 3π/4

Answer 3

You are correct, and I apologize for the mistake. The principal value for \(\cos^{-1}(x)\) is in the interval \([0, \pi]\). Since \(\cos\left(\frac{3\pi}{4}\right) = -\frac{1}{\sqrt{2}}\), the exact value of \(\cos^{-1}\left(-\sqrt{\frac{2}{2}}\right)\) is indeed \(\frac{3\pi}{4}\). Thank you for catching that.