Trigonometry/Derivative of Inverse Functions

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The inverse functions , etc. have derivatives that are purely algebraic functions.

If then and

.

So

Similarly,

.

If then and

.

So

If then and

.

So

Power seriesEdit

The above results provide an easy way to find the power series expansions of these functions.

 

This is uniformly convergent if   so can be integrated term by term. The constant of integration is zero since   , so

 
 

This is uniformly convergent if   so can be integrated term by term. The constant of integration is zero since   , so

 

Note that   has no power series expansion about   , as it is not defined for   and has an infinite derivative when   . An expansion about any point   in powers of   can be found uding Taylor's theorem; it will converge for   .