mjhjmtu

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:

documentclassarticle
usepackagetikz

tikzsethelp lines/.style=very thin
tikzsetMy Grid/.style=help lines,color=blue!50

begindocument
begintikzpicture
 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);
endtikzpicture
enddocument 

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

documentclassarticle
usepackagetikz
usetikzlibraryangles,quotes

tikzsethelp lines/.style=very thin
tikzsetMy Grid/.style=help lines,color=blue!50

begindocument
begintikzpicture
 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);

endtikzpicture
enddocument 

Here is an alternative to Hafid's nice answer, going further into the direction you started. You do not need to use any trigonometric function. Just using polar coordinates and projections is sufficient.

documentclassarticle
usepackagetikz
usetikzlibrarycalc,angles,quotes
tikzsethelp lines/.style=very thin
tikzsetMy Grid/.style=help lines,color=blue!50

begindocument
begintikzpicture
 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) coordinate (O) circle [radius=3cm];
 draw[red, very thick] (30:3cm) coordinate (A) 
 % (30:3cm) is a polar coordinate with angle 30 (degrees) and radius 3cm
 -- (0,0-|A) coordinate(Ax)
 % (0,0-|30:3cm) is a point that has the x coordinate of A and y=0
 % see https://tex.stackexchange.com/a/401429/121799 for more details
 node[midway,left]$sinalpha$;
 draw [very thick,orange] (3,0) -- (intersection cs:
 first line=(O)--(A),second line=(3,0)--(3,3)) coordinate(A')
 % (A') is at the intersections of the lines OA and the vertical line through (3,0)
 node[midway,right]$tanalpha$;
 pic[fill=green!50,angle radius=1cm,
 angle eccentricity=0.6, "$alpha$"] angle=Ax--O--A;
 % that's almost a 1-1 copy of what you can find on p. 560 of the manual 
 draw (O) -- (A');
 draw[very thick,blue] (O) -- (Ax) node[midway,below]$cosalpha$;
endtikzpicture
enddocument