How to embed plain text into math mode?

Probably you like to achieve this:

Instead of \text in first occasion you should use \parbox which enable to write two line text. After than you need to clean-up remedy with curly braces:

\begin{document} %starts document

\begin{align} %starts a formula block
\parbox{8.5em}{Number of looks in\\ azimuth direction}
    &= \frac{\text{Pixel spacing slant range}}
            {\text{Pixel spacing azimuth}}\times \sin (\text{Incidence angle})\\
    &= \frac{9.3685143}{3.5448059\times \sin (23.871^\circ)}
\end{align} %ends formula block

\end{document}

Addendum: Considering Andrew Swann comment, use siunitx for units and make text width a bit wider with geometry package I obtain:

\documentclass[a4paper,11pt]{article}
\usepackage[showframe]{geometry} % show page layout and make text width wider ...
                                 % in final use option "showframe" had to be removed
\usepackage{amsmath} % declares a predefined package for math
\usepackage{siunitx}

\begin{document} %starts document

\begin{align} %starts a formula block
\parbox{8.3em}{\centering Number of looks in\\ azimuth direction}
    &= \frac{\text{Pixel spacing slant range}}
            {\text{Pixel spacing azimuth}\times \sin (\text{Incidence angle})}\\[1ex]
    &= \frac{9.3685143}{3.5448059\times \sin (\SI{23.871}{\degree})}
\end{align} %ends formula block

\end{document}

Edit: It seems that factor \sin{...} had to be in the denominator. I correct this my misunderstanding of given MWE. Above mine MWE and image of equation is now corrected.

A slightly different approach (other than using \parbox), by applying the more general \genfrac command and using it for \textfrac.

According to Tobi's comment above, the equations provided by the O.P. contained an error regarding the position of the \sin.

I also used siunitx for the display of the angular value.

\documentclass[a4paper,11pt]{article}

\usepackage{siunitx}
\usepackage{amsmath} % declares a predefined package for math

\newcommand{\textfrac}[2]{\genfrac{}{}{0pt}{0}{\text{#1}}{\text{#2}}}
\begin{document} %starts document


\begin{align} %starts a formula block
  \textfrac{Number of looks}{in azimuth direction}
    &= \frac{\text{Pixel spacing slant range}}{\text{Pixel spacing azimuth}\times \sin \left(\text{Incidence angle}\right)}\\
    &= \frac{9.3685143}{3.5448059\times \sin (\SI{23.871}{\degree})}
\end{align} %ends formula block

\end{document} %ends document

You can't use \newline in \text. The two-line descriptor on the left-hand side can be done with tabular, so you don't have to guess a width. Here I use centering, you can choose left alignment.

\documentclass[a4paper,11pt]{article}
\usepackage{amsmath}

\begin{document}

\begin{align}
\begin{tabular}{@{}c@{}}
  Number of looks in\\
  azimuth direction
\end{tabular}
    &= \frac{\text{Pixel spacing slant range}}
            {\text{Pixel spacing azimuth}\times\sin (\text{Incidence angle})} \\
    &= \frac{9.3685143}{3.5448059\times \sin (23.871^\circ)}
\end{align}

\end{document}