In this applet you adjust the output range of inverse trigonometric relations so that only one [math]y[/math] value exist for each [math]x[/math] value. This makes them inverse trigonometric functions.

Select each trigonometric function on the left drop down. On the right side adjust the minimum and maximum allowed output values so there is only one output for each input. This range is shown with the brown curve. Note the corresponding domain of the function on the left is highlighted in brown to match.[br][br]Some ranges are better than other ranges, especially tangent and cotangent functions. Mathematicians have selected ranges for the inverse trigonometric functions that are very standard. These ranges are either [math]\left[-\frac{\pi}{2},\frac{\pi}{2}\right][/math] or [math]\left[0,\pi\right][/math]. Which functions work best with each range and why?[br][br]The inverse functions with standard output ranges are prefixed with "arc". For example, the inverse function of [math]\sin\left(x\right)[/math] may be written as [math]\sin^{-1}\left(x\right)\text{ , }\arcsin\left(x\right)\text{ or asin}\left(x\right)[/math]. Note that [math]\sin^{-1}\left(x\right)\ne\frac{1}{sin\left(x\right)}[/math] which may be easily misinterpreted since [math]\sin^2\left(x\right)=\left(\sin\left(x\right)\right)^2[/math].