Problems with TikZ calculations

Question

When can I use functions and do calculations in TikZ? How do I make this work:

\draw (0,0) arc(0:90:sqrt(15)); %not ok

why is this working:

\draw (0,0) arc(0:asin(1):5); %ok

with \usetikzlibrary{calc} is this:

 \draw (0,0) -- ($ (4,0) + sqrt(7)*(0,1) $);  %ok

the only way to do single coordinate calculations? why is this

 \draw (0,0) -- ($ (4,sqrt(7)) $);  %not ok

not working?

Minimal example:

\documentclass[tikz, border=1cm]{standalone}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\draw (0,0) arc(0:90:sqrt(15));              %not ok
\draw (0,0) arc(0:asin(1):5);                %ok
\draw (0,0) -- ($ (4,0) + sqrt(7)*(0,1) $);  %ok
\draw (0,0) -- ($ (4,sqrt(7)) $);            %not ok
\end{tikzpicture}
\end{document}

Answer 1

You need to wrap the expression into { } to hide the second pair of ( ) from the TeX parser. Without the { } a ( will be closed by the next ) even if it belongs to another (. This means arc(0:90:sqrt(15)) will be taken as arc(0:90:sqrt(15) without the second ). This causes basically two errors, one in the expression because it misses the ) and another one in the \draw path which doesn't know what to do with a single ). With the { } it works because any { must be closed first with a } before the ) is taken.

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\begin{tikzpicture}
    \draw (0,0) arc(0:90:{sqrt(15)}); %now ok
    \draw (0,0) arc(0:{asin(1)}:5); %ok
    \draw (0,0) -- ($ (4,0) + sqrt(7)*(0,1) $);  %ok
    \draw (0,0) -- ($ (4,{sqrt(7)}) $);  %now ok
\end{tikzpicture}

\end{document}

Answer 2

Edit: This is incorrect, disregard this.. Well, it's correct up to the point that the coordinates can't have math in them. Disregard it anyway.

It's just how the Tikz calc library is set up. If we have a look at section 13.5 of the manual

The general syntax is the following:
([ options ]$ coordinate computation $).
As you can see, the syntax uses the TEX math symbol $ to indicate that a “mathematical computation” is involved. However, the $ has no other effect, in particular, no mathematical text is typeset.
The coordinate computation has the following structure:
1. It starts with < factor > * < coordinate > < modifiers >
2. This is optionally followed by + or - and then another < factor > * < coordinate > < modifiers >
3. This is once more followed by + or - and another of the above modified coordinate; and so on.
In the following, the syntax of factors and of the different modifiers is explained in detail.

The following you will have to read in the manual for yourself. The important thing here is that the math can only occur in the factor and the modifiers. Not in the coordinate itself, which explains why your second example does not work. The reason the first example doesn't work is explained in the manual as well. It is the same type of restriction in the Tikz grammar.