calculator's angle answer for trig ratios that can work in more than 1 quadrant on the unit circle

1. Why does the calculator do a cc (counterclockwise) rotation for positive trig ratios instead of clockwise,

2. and a clockwise rotation for negative sine & tan instead of cc

3. and a counterclockwise rotation for negative cos ratios instead of a clockwise

ie. in degree mode

$\cos^{-1}(-5/12)=114.62$

$\sin^{-1}(-5/12)=-24.62$

$\tan^{-1}(-5/12)=-22.61$

Is it maybe picking the value that involves the least amount of computing power? or is it a matter of convention? or am I overlooking something?

This actually has to do with the way inverse trig functions are defined. For a function to be invertible there must be one input for every output. Graphically, this is equivalent to passing the horizontal line test. Now, trig functions are periodic and as such are very much not invertible. The way we get around this is to restrict the domain of each function to a region that passes the horizontal line test.

For $\sin(x)$ the region that we take is $-\frac{\pi}{2}\leq x \leq \frac{\pi}{2}$, or $-90^{\circ} \leq x \leq 90^{\circ}$ in degree mode, as seen in the following plot:

For $\cos(x)$ the region we take is $0\leq x \leq \pi$, or $0^{\circ} \leq x \leq 180^{\circ}$ in degree mode. Note that we could also have taken $-\pi \leq x \leq 0$, but for convenience we take $x$ to be a positive angle.

Lastly, for $\tan(x)$ we can take a full period around the origin, so $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$, or $-90^{\circ} \leq x \leq 90^{\circ}$ in degree mode.