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Powers of trigonometric functions

2021-07-02T12:11:56Z

SparksMaths:

[[Category:Ambiguities]]
[[Category:Inconsistencies]]
[[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:

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

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 <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.

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

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==

<references/>

Powers of trigonometric functions

2021-07-02T12:10:35Z

SparksMaths:

[[Category:Ambiguities]]
[[Category:Inconsistencies]]
[[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:

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

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 <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.

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

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==

<references/>

Powers of trigonometric functions

2021-07-02T12:07:49Z

SparksMaths: Added discussion of inverse trig functions and their index notation

[[Category:Ambiguities]]
[[Category:Inconsistencies]]
[[Category:Unpleasantness]]

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 <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.

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

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==

<references/>