Question

Find the exact real value of arccos: The quantity negative square root of two divided by two.

Answer 1

We want to find the value of cos⁻¹ x, where $x = - \frac{ \sqrt{2} }{2}$

Note that
cos(45°) = sin(45°) = 1/√2;  tan(45°) = 1.
Also,
45° = π/4 radians, and
√2/2 = 1/√2

The cosine function is negative in quadrants 2 and 3.

Therefore
cos⁻¹ (-√2/2) = π - π/4 radians = (3/4)π radians or 
                      = π + π/4 radins  = (5/4)  radians
In degrees,
cos (-√2/2) = 135° or 225°

Answer:
 $\frac{3 \pi }{4}\,or\, \frac{5 \pi }{4}\,radians$
In degrees, it is 135° or 225°.