# 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?

• Conventionally, counter clockwise rotations are described by positive angles. But it looks like your question is more about the ranges of the inverse trigonometric functions. – John Doe Apr 22 at 0:25
• Try using Mathjax: Surround your formulas with $signs, use \ before a trig function, and {} between the start and end of a superscript. E.g. \$\cos^{-1}(-5/12)=114.62\\$ – man and laptop Apr 22 at 0:28
• This tutorial explains how to typeset mathematics on this site. – N. F. Taussig Apr 22 at 1:13

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.