Bdtyk

I am learning TikZ with the pgfmanual. The example given is as follows:

I have achieved the following so far:

The code is as follows:

documentclass{article}

usepackage{tikz}



tikzset{help lines/.style=very thin}

tikzset{My Grid/.style={help lines,color=blue!50}}



begin{document}

begin{tikzpicture}

  draw[My Grid] (-4,-4) grid (4,4);

  draw (-5,0) node[left] {$(-5,0)$} -- (5,0) node[right] {$(5,0)$};

  draw (0,-5) node[below] {$(0,-5)$} -- (0,5) node[above] {$(0,5)$};

  draw (0,0)  circle [radius=3cm];

  shadedraw[left color=gray, right color=green, draw=green!50!black] (0,0) -- (0.75,0)  arc [start angle=0, end angle=30, radius=1cm] -- cycle;

  draw[red, very thick] (30:3cm) -- (2.6,0);

  draw [very thick,orange] (3,0) -- (3,1.7);

end{tikzpicture}

end{document} 

To achieve the intersection of the slope and tangent, pgfmanual uses the concept of path and intersections library which is very confusing.

Is there an easier way to tell the system to draw a line from point A to sin(30) as point B and to draw tangent of the angle proportionately instead of using direct numbers.

Please suggest alternative apart from intersections, polar coordinates and paths. Because the pgfmanual already uses them which is hard to understand.

Since in cos(30) and tan(30) there are parentheses you must put these functions inside curly brackets {}

documentclass{article}

usepackage{tikz}

usetikzlibrary{angles,quotes}



tikzset{help lines/.style=very thin}

tikzset{My Grid/.style={help lines,color=blue!50}}



begin{document}

begin{tikzpicture}

  draw[My Grid] (-4,-4) grid (4,4);

  draw (-5,0) node[left] {$(-5,0)$} -- (5,0) node[right] {$(5,0)$};

  draw (0,-5) node[below] {$(0,-5)$} -- (0,5) node[above] {$(0,5)$};

  draw (0,0)  circle [radius=3cm];



%   shadedraw[left color=gray, right color=green, draw=green!50!black]

% (0,0) -- (0.75,0)  arc [start angle=0, end angle=30, radius=0.75cm] --  cycle;



  coordinate(O)at(0,0);

  draw[red, very thick] (30:3cm)coordinate(A) 

                         --({3*cos(30)},0)coordinate(B);



  draw [very thick,orange] (3,0) -- (3,{3*tan(30)})coordinate(C);



  pic[fill=green!50!black,

       angle radius=0.75cm,

       angle eccentricity=1.2,

       "(alpha)"] {angle=B--O--A};



   draw (O)--(C);



end{tikzpicture}

end{document}