When I type Trig functions using {} or () I seem to get the same result, what is the difference between the two?
[Tex/LaTex] Using Trigonometric Functions
math-modemath-operators
Related Solutions
There are in fact four important differences:
\left( ... \right)etc. scales according to the height and depth of its contents. This scaling is "dumb" in the sense that it will always take the full height and depth into account (how much of the expression is covered is controlled by\delimitershortfalland\delimiterfactor): for example, in\left( \rule{1cm}{1cm} \right), the parentheses reach far below the square. A more relevant example is\left( \sum_a^b \right)where the parentheses also cover the sum limits. The simple delimiters(and)and also the manually-sized delimiters\big(etc. don't scale.\left ... \rightforms a group: if you say\newlength\mylength \[ \left( \mylength=1cm \right) \the\mylength \]you get0.0ptbecause the value was reset. More importantly, you cannot have line breaks inside\left ... \rightgroups, neither manual nor automatic ones, without special trickery. Any\leftneeds a matching\right.- Some characters produce different glyphs when being applied to
\leftetc. For example,<produces a less-than sign, while\left<produces an angle bracket.\bigetc. use the same interpretation as\left(because they use\leftinternally). Technically,\leftuses the delimiter code, while unadorned characters use the mathematical code. - The spacing is different. Technically,
\left ... \rightinserts an inner node, while(inserts an opening node. This becomes visible in$\sin()$vs.$\sin\left(\right). Therefore you can never simply replace(by\left(and vice versa, you always have to check whether the spacing comes out right. An automatic solution to this issue is offered in Spacing around\leftand\right, but the spacing within\left...\rightcan still be different as explained in this answer.
After spending some time looking for this, I found this post that suggested defining the new commands for the omitted inverse trig functions.
Here I've augmented that with the full suit of hyperbolic and inverse hyperbolic functions for convenience, as google doesn't turn anything up for this search, nor does the other post come up if one is searching for the inverse hyperbolic functions, specifically.
\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\sech}{sech}
\DeclareMathOperator{\csch}{csch}
\DeclareMathOperator{\arcsec}{arcsec}
\DeclareMathOperator{\arccot}{arcCot}
\DeclareMathOperator{\arccsc}{arcCsc}
\DeclareMathOperator{\arccosh}{arcCosh}
\DeclareMathOperator{\arcsinh}{arcsinh}
\DeclareMathOperator{\arctanh}{arctanh}
\DeclareMathOperator{\arcsech}{arcsech}
\DeclareMathOperator{\arccsch}{arcCsch}
\DeclareMathOperator{\arccoth}{arcCoth}
\begin{document}
\[
\sech x \cschx \arcsec x \arccot x \arccsc x \arccosh x \arcsinh x \arctanh x \arcsech x arccsch x \arccoth x
\]
\end{document}
Best Answer
They cannot be the same as follows.