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<math display="block"> \operatorname{atan2}(y, x) = \lim_{z \to x^+}\arctan\left(\frac{y}{z}\right) + \frac{\pi}2\sgn(y)\sgn(x)\left(\sgn(x)-1\right) </math> |
<math display="block"> \operatorname{atan2}(y, x) = \lim_{z \to x^+}\arctan\left(\frac{y}{z}\right) + \frac{\pi}2\sgn(y)\sgn(x)\left(\sgn(x)-1\right) </math> |
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, that is useful e.g. to define the [[Multivalued_function|multivalued]] version of each inverse trigonometric function: e.g. <math>\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\}</math>. However, this might appear to conflict logically with the common semantics for expressions such as {{math|sin<sup>2</sup>(''x'')}} (although only {{math|sin<sup>2</sup> ''x''}}, without brackets, is the really common one) [...] This potentially avoids confusion with the multiplicative inverse, which should be represented by {{math|sin<sup>−1</sup>(''x'')}}, {{math|cos<sup>−1</sup>(''x'')}}, etc., or, better, by {{math|sin<sup>−1</sup> ''x''}}, {{math|cos<sup>−1</sup> ''x''}}, etc. |
, that is useful e.g. to define the [[Multivalued_function|multivalued]] version of each inverse trigonometric function: e.g. <math>\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\}</math>. However, this might appear to conflict logically with the common semantics for expressions such as {{math|sin<sup>2</sup>(''x'')}} (although only {{math|sin<sup>2</sup> ''x''}}, without brackets, is the really common one) [...] Another convention used by a few authors is to use an [[UPPERCASE|uppercase]] first letter, along with a {{math|<sup>−1</sup>}} superscript: {{math|Sin<sup>−1</sup>(''x'')}}, {{math|Cos<sup>−1</sup>(''x'')}}, {{math|Tan<sup>−1</sup>(''x'')}}, etc.<ref name="Bhatti_1999" /> This potentially avoids confusion with the multiplicative inverse, which should be represented by {{math|sin<sup>−1</sup>(''x'')}}, {{math|cos<sup>−1</sup>(''x'')}}, etc., or, better, by {{math|sin<sup>−1</sup> ''x''}}, {{math|cos<sup>−1</sup> ''x''}}, etc. |
Revision as of 19:37, 25 January 2022
Formula without apparent conditional construct:
, that is useful e.g. to define the multivalued version of each inverse trigonometric function: e.g. . However, this might appear to conflict logically with the common semantics for expressions such as sin2(x) (although only sin2 x, without brackets, is the really common one) [...] Another convention used by a few authors is to use an uppercase first letter, along with a −1 superscript: Sin−1(x), Cos−1(x), Tan−1(x), etc.[1] This potentially avoids confusion with the multiplicative inverse, which should be represented by sin−1(x), cos−1(x), etc., or, better, by sin−1 x, cos−1 x, etc.