Powers of trigonometric functions

From Why start at x, y, z

When writing a power of a trigonometric function or logarithm, it's common to write the power as a superscript before the brackets:

\[ \sin^2(x) = (\sin(x))^2 \]

This conflicts with the conventional notation for inverses and repeated application of functions:

\[ f^2(x) = f(f(x)) \]

\[ y = f^{-1}(x) \implies x = f(y) \]

In his entry on Notation in the Edinburgh Encyclopedia [1], Charles Babbage is scathing about this convention.

Although a definition can not be false, it may be improper.

Further aggravating this issue is the commonplace use of conventional function notation to denote the inverse of trigonometric functions. When used alongside the \(\sin^2(x)\) notation this juxtaposes the contradictory notation quite sharply, and may cause potential confusion for learners. For example:

\[\sin^2(x) = (\sin(x))^2 \] (index notation, as in the original example)

but \[ \sin^{-1}(x) \neq \frac{1}{\sin(x)}\] rather \[ \sin^{-1}(x) = \arcsin(x) \] (function notation)

It is arguable that conventions such as \(\arcsin(x)\) for inverse sine, and \(\csc(x)\) for the reciprocal of sine (or often \(\textrm{cosec}(x)\) - at least in UK schools), are better used to avoid these notational contradictions.


  1. The Edinburgh Encyclopedia, 1830, pp. 398-399. Copy on HathiTrust. Transcribed copy.