Powers of trigonometric functions: Difference between revisions

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[[Category:Inconsistencies]]
[[Category:Inconsistencies]]
[[Category:Unpleasantness]]
[[Category:Unpleasantness]]
[[Category:Conflicting definitions]]


When writing a power of a trigonometric function or logarithm, it's common to write the power as a superscript before the brackets:
When writing a power of a trigonometric function or logarithm, it's common to write the power as a superscript before the brackets:
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\[ y = f^{-1}(x) \implies x = f(y) \]
\[ y = f^{-1}(x) \implies x = f(y) \]


In his entry on Notation in the Edinburgh Encyclopedia <ref>The Edinburgh Encyclopedia, 1830, pp. 398-399. [https://babel.hathitrust.org/cgi/pt?id=chi.21708663&view=1up&seq=431 Copy on HathiTrust]</ref>, Charles Babbage is scathing about this convention.  
In his entry on Notation in the Edinburgh Encyclopedia <ref>The Edinburgh Encyclopedia, 1830, pp. 398-399. [https://babel.hathitrust.org/cgi/pt?id=chi.21708663&view=1up&seq=431 Copy on HathiTrust]. [https://www.maths.ed.ac.uk/~csangwin/Babbage1830.pdf Transcribed copy].</ref>, Charles Babbage is scathing about this convention.  


<blockquote>Although a definition can not be false, it may be improper.</blockquote>
<blockquote>Although a definition can not be false, it may be improper.</blockquote>

Latest revision as of 14:50, 6 July 2021


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.

References

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