Scale math font with newtxsf

Question

I'm using uarial as text font and want to use newtxsf as a nearly matching math font. I scaled uarial to match the height of latin modern as described in this post: https://tex.stackexchange.com/a/206677.

Still I'm encountering two problems:

• I don't know how to scale newtxsf correctly, since \the\ht0 stays the same when I change the scaled option. If there is no way to display the height directly I would try some values and choose the one that fits the most.
• in math mode, "sin", "cos", ... are not set in uarial but in latin modern. Is there a way to tell newtxsf to use \sffamily instead of \rmfamily? I tried to solve this using the newtxsf documentation but failed. Or is there another package that matched uarial?

My Code so far

\documentclass[fontsize=12pt, paper=a4, parskip=half]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[scale=0.93]{tgheros}
\usepackage[scaled=1]{newtxsf}
 
\renewcommand*{\familydefault}{\sfdefault}

\begin{document}
    \textbf{Displaying height of fonts}\par
    \sbox0{\sffamily A}heros: \the\ht0 \\
    \sbox0{\rmfamily A}latin modern: \the\ht0\\
    \sbox0{\ttfamily A}newtxsf: \the\ht0\par
    \textbf{Math mode}  $ Q = P\tan(\varphi)$
\end{document}

Compiled Document:

Answer 1

My suggestion is to try with no scaling; in this case we get the height of a sans serif A is 8.87999pt and the height of a math A is 8.148pt.

Therefore the scaling factor is 8.148/8.87999=0.91757 (rounded to the fifth decimal digit). Now with

\documentclass[fontsize=12pt, paper=a4]{scrreprt}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage[scale=.91757]{tgheros}
\usepackage[scaled=1]{newtxsf}

\renewcommand*{\familydefault}{\sfdefault}
\renewcommand*{\rmdefault}{\sfdefault}

% this to get the operators in the sans serif font
\DeclareSymbolFont{operators}{\encodingdefault}{\sfdefault}{m}{n}

\begin{document}

    \textbf{Displaying height of fonts}

    \sbox0{\sffamily A}heros: \the\ht0

    \sbox0{$A$}newtxsf: \the\ht0

    \textbf{Math mode}  $ Q = P\tan(\varphi)$

\end{document}

Answer 2

Note that, in unicode-math, the solution is Scale=MatchLowercase, or sometimes, Scale=MatchUppercase. You can even set it as \defaultfontfeatures.

It also sets the main font as both the operator font and the \mathrm font, but you can override this, with \setoperatorfont, \setmathrm, etc.