Question

How do you find the exact value of `cos^-1 (-sqrt2/2)`?

Answer 1

`cos^(-1)(-sqrt(2)/2) = 135^o or 225^o`
`color(white)("XXXX")`if we restrict the range to `[0,360^o)`

Explanation:

`(-sqrt(2)/2) = (-1/sqrt(2))`

If `cos(1/sqrt(2)) = theta` then `cos(theta) = (-1/sqrt(2))`

If `theta epsilon [0,2pi)`
we have the two possibilities indicated in the diagram below:

with `theta` (measuring from the positive x-axis) as either
`180^o - 45^o = 135^o`
or
`180^o + 45^o = 225^o`

For people who prefer their angles in radians that is
`pi - pi/4 = (3pi)/4`
or
`pi+pi/4 = (5pi)/4`