This is a possible solution. The point c are on the ray of radius found by the intersections skill from intersections
library. Based on \tan \alpha
expression, the 2nd line segment has length 1 on the x-axis which is inside the circle of radius 4.
Update Further automation is attempted where a macro \ang
can be set between 0 and 90 degree, the intersections library will find the intersection points autmatically. The demonstration is for \ang=45
.
- How do I label the drawn radius "r"? use node[pos=xx, above=1pt] in the radius
- Two line segments are to be drawn that are perpendicular to the x-axis. Use intersections of two curve/lines to find the intersection points b anc c.
- How do I put a dot at the intersection of the second line segment and the ray containing the drawn radius? Use of
\node[dot](<internal label>){};

Code
\documentclass[border=10pt]{standalone}%
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{calc,positioning,intersections}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\def\ang{45} % 0< \ang<90
\begin{document}
%\begin{center}
\begin{tikzpicture}[
dot/.style={
fill,
circle,
inner sep=2pt
}
]
\clip (-1,-1) rectangle (5.5,5.5);
\draw[dashed,fill=white,name path=curve] (0,0) coordinate(O){} circle [radius=4];
\draw[<->] (-5,0) -- (5,0) node[below] {$x$};
\draw[<->] (0,-5) node (yaxis) {} -- (0,5) node[right] {$y$};
\path[name path=lineb] (0,0) -- (\ang:5);
\draw[name intersections={of=curve and lineb, by={b}},thick]{};
\node[dot,label={left:$(r\cos\alpha, \, r\sin\alpha)$}] at (b) {};
\draw[->] (0,0) --node[pos=0.7,above]{$\alpha$} (0.8,0) arc (0:\ang:0.8cm) ;
\draw (O) --node[pos=0.7,above=1pt]{$r$} (b) -- (b |- O);
\path[name path=linec] (1,0) -- (1,3);
\draw[name intersections={of=lineb and linec, by={c}},thick]{};
\draw[] (c) --node[midway,right](){$\tan \alpha$} (c |- O);
\node[dot] at (c){};
\end{tikzpicture}
%\end{center}
\end{document}

Code
\documentclass[border=10pt]{standalone}%{article}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{calc,positioning,intersections}
\usepackage{pgfplots}
\begin{document}
%\begin{center}
\begin{tikzpicture}[
dot/.style={
fill,
circle,
inner sep=2pt
}
]
\clip (-0.5,-0.5) rectangle (7,7);
\draw[dashed,fill=white] (0,0) node(O){} circle [radius=4];
\draw[<->] (-5,0) -- (5,0) node[below] {$x$};
\draw[<->] (0,-5) node (yaxis) {} -- (0,5) node[right] {$y$};
\node[dot,label={left:$(r\cos\alpha, \, r\sin\alpha)$}] at (3.464101615,2) (a) {};
\draw (0,0) -- (3.464101615,2);
\draw[->] (0,0) --node[pos=0.7,above]{$\alpha$} (0.8,0) arc (0:30:0.8cm) ;
\draw (a) -- (a |- O);
% tan alpha
\draw[dashed,name path=linea] (O) -- (a);
\path[name path=linec] (1,0) -- (1,4);
\path[name intersections={of=linea and linec, by={c}},thick]{};
\draw[] (c) --node[midway,right](){$\tan \alpha$} (c |- O);
\node[dot] at (c){};
\end{tikzpicture}
%\end{center}
\end{document}
pgfplots
manuals are by far the best manuals out there.