The inverse trigonometric (or cyclometric or arcus) functions — arccosine, arcsine, arctangent, arcsecant, arccosecant, and arccotangent — are the inverses of the trigonometric functions. They each take a ratio/scaling-factor as input and return the corresponding angle/arclength as output. E.g. since \(\cos(45°) = \frac{\sqrt{2}}{2}\) we also have \(\operatorname{arccos}\bigl(\frac{\sqrt{2}}{2}\bigr) = 45°.\) Sometimes, in US textbooks or on calculators, they are instead denoted with a little \(-1\) superscript. E.g. \(\cos^{-1}(x)\) is the same thing as \(\operatorname{arccos}(x),\) and corresponds to the button labelled cos-1 on most calculators.
The domain of each of these inverse functions are the same as the ranges of their corresponding trigonometric function. However, because each trigonometric function is periodic, specifically not one-to-one, their range must each be restricted to the principal branch of the domain of the corresponding trigonometric function.
| \(\operatorname{arccos}\) | \(\operatorname{arcsin}\) | \(\operatorname{arctan}\) | \(\operatorname{arcsec}\) | \(\operatorname{arccsc}\) | \(\operatorname{arccot}\) | |
|---|---|---|---|---|---|---|
| Domain | \([-1,1]\) | \([-1,1]\) | \((-\infty,\infty)\) | \((-\infty,-1]\cup[1,\infty)\) | \((-\infty,-1]\cup[1,\infty)\) | \((-\infty,\infty)\) |
| Range | \([0°, 180°]\) | \([-90°, 90°]\) | \((-90°, 90°)\) | \([0°, 90°)\cup(90°, 180°]\) | \([-90°,0)\cup(0,90°]\) | \((0°, 180°)\) |
Technically speaking, because of that range restriction,
each of these function isn’t really an inverse
but only a right inverse (or a section)
of their corresponding trigonometric function.
I.e. it will always be the case that \(\cos\bigl(\operatorname{arccos}(x)\bigr) = x,\)
but \(\operatorname{arccos}\bigl(\cos(\theta)\bigr) = \theta\)
only for \(\theta\) in cosine’s principle branch between 0° and 180°.
In practice, we usually don’t have to think about this,
but it’s important to keep in the back of our mind.
When asked What angles correspond to an \(x\)-coordinate of ¾ on the unit circle?
we should know that there are infinitely many such angles;
even though we only get the one value \(\operatorname{arccos}(¾) ≈ 41.41°\) from arccosine,
we should know that \(-41.41°\) and \(401.41°\) and \(318.59°\) and so on,
are also such angles.