It's a simple question but I am a little confused. The value of $cos^{-1} (-0.5)$ , is it 2.0943 or 120 ?
[Math] calculator issue: radians or degrees for inverse trig
inversetrigonometry
Related Solutions
Note that $$\cos{30^{\circ}} = \cos{\frac{\pi}{6}} = \frac{\sqrt{3}}{2} \approx 0.866025$$
since we can convert from degrees into radians by using
$$1^{\circ} = \frac{\pi}{180}$$
A decent "mnemonic" to remeber some basic trig function values is the following table:
$$\begin{array}{lr} t & \sin{t} \\ 0 & \frac{\sqrt{0}}{1} \\ \frac{\pi}{6} & \frac{\sqrt{1}}{1} \\ \frac{\pi}{4} & \frac{\sqrt{2}}{2} \\ \frac{\pi}{3} & \frac{\sqrt{3}}{2} \\ \frac{\pi}{2} & \frac{\sqrt{4}}{2} \end{array}$$
Then $\cos{t}$ can be found from the relation $\cos^2 t + \sin^2 t = 1$.
Not an answer, but a discussion of the closeness of the situation.
Since $53^\circ$ and $38^\circ$ are very nearly complementary, we have that $\sec 38^\circ \approx \csc 53^\circ$ ... with the left-hand side being ever-so-slightly larger than the right-hand side.
As the first diagram suggests, for big enough (first-quadrant) angles $\theta$, we have that $\tan\theta$ exceeds $\csc\theta$; and, according to that first diagram, $53^\circ$ seems to be one of those "big enough" angles ... but just barely. Is it big enough that the $\tan 53^\circ$ also exceeds the slightly-larger value, $\sec 38^\circ$? Well, the middle diagram confirms that it is (though again: just barely), but of course having a computer program draw an accurate diagram is really no better than using a calculator compute the values.
What makes the approximations especially-tricky here is that $53^\circ$ is very close to the magic (or, should I say, "golden"?) angle, $\theta_\star = 51.8...^\circ$, marking the threshold of those "big enough" angles. If the problem had been to compare, say, $\tan 70^\circ$ with $\sec 21^\circ$, then we would have had more confidence in our ability to fiddle with the numbers.
All things considered, this seems like a bad exercise for a test. I wonder if there was an error in the test question.
Best Answer
If you think of $\cos^{-1}(x)$ as an angle, then you can express the answer in either degrees or radians.
If you think of this as a real number, though, then it always takes on a value between 0 and $\pi$ (which is numerically equal to the angle in radians).