Question

How do you evaluate `sec^-1(-2)` without a calculator?

Answer 1

`x = \pm (2\pi)/3`.

Explanation:

What do you mean exactly with that exponent `-1`?

If you mean the power, we have

`sec(x)=1/cos(x) \implies sec^{-1}(x) = 1/(1/cos(x))=cos(x)`

Thus, `sec^{-1}(-2)=cos(-2)`, and I see no way to calculate it manually.

If you mean the inverse function, we want to find a number `x` such that `sec(x)=-2`, or if you prefer, `1/cos(x)=-2`

This clearly leads to `cos(x)=-1/2`, and it is a known angle: `x = \pm (2\pi)/3`, depending on which interval you choose to invert the cosine function.