Evaluating $\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$ and $\sec^{-1}\left(-\sqrt{2}\right)$. Why are my answers ($5\pi/6$ and $3\pi/4$) incorrect?

Question

$$\sec^{-1}\left(\frac{-2}{\sqrt{3}}\right)$$

$$\sec^{-1}\left(-\sqrt{2}\right)$$

I know that they are set up like $\sec(y)=x$ ...

$$\sec(y)=\frac{-2}{\sqrt{3}}$$

$$\sec(y)=-\sqrt{2}$$

I got $\frac{5\pi}{6}$ for the first one and $\frac{3\pi}{4}$ for the second, but these are wrong and I'm not sure why.

The only feedback I got was "An inverse trigonometric function takes a numeric value as input, and returns an angle as the output."

Answer 1

Perhaps the answer is looking for degrees in which case, you have 150$^\circ$ for the first one and 135$^\circ$ for the second one.

Answer 2

The restricted domain for $\sec$ is not consistently defined. Some texts use;

$$y=\sec^{-1} x \iff \sec y = x \mbox{ and } y\in [0,\pi/2)\cup [\pi, 3\pi/2). $$

So perhaps that is your snag. The reason for choosing this domain is that it makes the derivative formula nicer.