# A drawing for a trigonometric Identity (using the let-in syntax.)

I have the display for showing the identities for the sine and cosine of a difference of two numbers. It is drawn exactly as I want. I want to know why the following two commands do not draw the same line segment at the midpoint on an arc of a circle centered at Q. (The vector difference (Q)-(P) is in the first quadrant, and (Q)-(R) is in the second quadrant. Using atan to determine the angle of the vector (Q)-(R) requires one to add 180.) I give two commands for drawing this line segment. The one drawn in blue is the one I want, and the one drawn in green is misplaced. Why is the green line segment centered at Q? How do I change the syntax so that it is drawn where the blue line segment is drawn?

Here is the command that draws the blue line segment.

\draw[draw=blue,fill] let \p1=($(Q)-(R)$), \n1={atan(\y1/\x1)}, \p2=($(Q)-(P)$), \n2={atan(\y2/\x2)} in
($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!-3pt!(Q)$) -- ($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!3pt!(Q)$);


Here is the command that draws the green line segment.

\draw[draw=green,fill] let \p1=($(Q)-(R)$), \n1={atan(\y1/\x1)}, \p2=($(Q)-(P)$), \n2={atan(\y2/\x2)} in
($(Q) +({0.5*(\n1+\n2+180)}:{-0.5-5pt})$) -- ($(Q) +({0.5*(\n1+\n2+180)}:{-0.5+5pt})$);


I understand that there the angles package is designed for marking angles. I am using this as an opportunity to get familiar with the let - in syntax.

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}

\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,decorations.pathreplacing}

\begin{document}

\noindent \hspace*{\fill}
\begin{tikzpicture}

\coordinate (O) at (0,0);

\draw[-latex] (O) -- (10:3);
\coordinate (label_for_ray_r_1) at ($(10:3) +(10:3mm)$);
\node at (label_for_ray_r_1){$r_{1}$};
\draw[-latex,dashed] (O) -- (190:7);
\draw[-latex,name path=ray_2] (O) -- (130:9);
\coordinate (label_for_ray_r_2) at ($(130:9) +(130:0.3)$);
\node at (label_for_ray_r_2){$r_{2}$};
\draw[-latex,name path=ray_3] (O) -- (150:8);
\coordinate (label_for_ray_r_3) at ($(150:8) +(150:0.3)$);
\node at (label_for_ray_r_3){$r_{3}$};

\coordinate (Q) at (130:7.5);
\draw[fill] (Q) circle (1.5pt);
\coordinate (P) at ($(O)!(Q)!(150:8)$);
\draw[fill,blue] (P) circle (1.5pt);
\draw[name path=path_PQ] (P) -- (Q);

%A right-angle mark is drawn at P.
\coordinate (U) at ($(P)!4mm!45:(O)$);
\draw[dashed] (U) -- ($(P)!(U)!(O)$);
\draw[dashed] (U) -- ($(P)!(U)!(Q)$);

\coordinate (A) at ($(O)!(Q)!(190:4)$);
\coordinate (R) at ($(A)!(P)!(Q)$);
\draw[fill,blue] (R) circle (1.5pt);
\draw (P) -- (R);
\draw (Q) -- (R);
\draw (R) -- (A);

%A right-angle mark is drawn at R.
\coordinate (U_4) at ($(R)!4mm!45:(P)$);
\draw[dashed] (U_4) -- ($(R)!(U_4)!(Q)$);
\draw[dashed] (U_4) -- ($(R)!(U_4)!(P)$);

%The label for O is typeset. An invisible path is drawn 3mm below the line containing
%ray_1. Since the angle for y is between rays inclined at 130 degrees and 150 degrees,
%the label for O is centered along a ray inclined at an angle of -40 degrees.
\coordinate (label_for_O) at ($(0,0)!3mm!-90:(10:1)$);
\node at (label_for_O){$O$};

%The label for Q is typeset.
\coordinate (label_Q_left) at ($(Q)!-1cm!(A)$);
\coordinate (label_Q_right) at ($(Q)!-1cm!(P)$);
\coordinate (label_Q) at ($(label_Q_left)!0.5!(label_Q_right)$);
\node[blue] at ($(Q)!3mm!(label_Q)$){$Q$};

%The label for P is typeset.
\coordinate (label_P_above) at ($(P)!-15mm!(Q)$);
\coordinate (label_P_below) at ($(P)!-15mm!(R)$);
\coordinate (label_P) at ($(label_P_above)!0.5!(label_P_below)$);
\node[blue] at ($(P)!3mm!(label_P)$){$P$};

%The label for R is typeset.
\coordinate (label_R) at ($(R)!-4mm!(P)$);
\node[blue] at (label_R){$R$};

%The angle at O with a measure of 180 - x is drawn. The angle is marked with "|".
\draw[draw=blue] (O) ++(190:0.75) arc (190:150:0.75);
\draw[draw=blue] ($(170:0.75) +(170:-3pt)$) -- ($(170:0.75) +(170:3pt)$);

%The angle at O with a measure of y is drawn.
\draw[draw=blue] (O) ++(150:0.9) arc (150:130:0.9);
\coordinate (label_for_y) at (140:1.1);
\node[font=\footnotesize] at (label_for_y){$y$};

%An angle at P with measure x is drawn. The angle is marked with "|".
\draw[draw=blue] let \p1=($(R)-(P)$), \n1={atan(\y1/\x1)} in ($(P)!0.5cm!(O)$) arc (-30:\n1:0.5);
\draw[draw=blue] let \p1=($(R)-(P)$), \n1={atan(\y1/\x1)} in ($($(P) +({0.5*(\n1-30)}:0.5)$)!-3pt!(P)$) -- ($($(P) +({0.5*(\n1-30)}:0.5)$)!3pt!(P)$);

%An angle at Q with measure x is drawn. The angle is marked with "|".
\draw[draw=blue] let \p1=($(Q)-(R)$), \n1={atan(\y1/\x1)}, \p2=($(Q)-(P)$), \n2={atan(\y2/\x2)} in ($(Q)!0.5cm!(R)$) arc (\n1:{\n2 - 180}:0.5);
\draw[draw=blue,fill] let \p1=($(Q)-(R)$), \n1={atan(\y1/\x1)}, \p2=($(Q)-(P)$), \n2={atan(\y2/\x2)} in
($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!-3pt!(Q)$) -- ($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!3pt!(Q)$);

\draw[draw=green,fill] let \p1=($(Q)-(R)$), \n1={atan(\y1/\x1)}, \p2=($(Q)-(P)$), \n2={atan(\y2/\x2)} in
($(Q) +({0.5*(\n1+\n2+180)}:{-0.5-5pt})$) -- ($(Q) +({0.5*(\n1+\n2+180)}:{-0.5+5pt})$);

\end{tikzpicture}

\end{document}


It's a problem of mixing units. Using simply ($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5-5pt})$) PGF assumes unit is pt so it simply moves you -5.5pt away from Q (similarly, ($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5+5pt})$) moves you just 4.5pt away from Q. What you need is to specify cm for the 0.5

\draw[draw=green,fill]
let
\p1=($(Q)-(R)$),
\n1={atan(\y1/\x1)},
\p2=($(Q)-(P)$),
\n2={atan(\y2/\x2)}
in
($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5cm-5pt})$) --
($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5cm+5pt})$);


The complete code:

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}

\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,decorations.pathreplacing}

\begin{document}

\noindent \hspace*{\fill}
\begin{tikzpicture}

\coordinate (O) at (0,0);

\draw[-latex] (O) -- (10:3);
\coordinate (label_for_ray_r_1) at ($(10:3) +(10:3mm)$);
\node at (label_for_ray_r_1){$r_{1}$};
\draw[-latex,dashed] (O) -- (190:7);
\draw[-latex,name path=ray_2] (O) -- (130:9);
\coordinate (label_for_ray_r_2) at ($(130:9) +(130:0.3)$);
\node at (label_for_ray_r_2){$r_{2}$};
\draw[-latex,name path=ray_3] (O) -- (150:8);
\coordinate (label_for_ray_r_3) at ($(150:8) +(150:0.3)$);
\node at (label_for_ray_r_3){$r_{3}$};

\coordinate (Q) at (130:7.5);
\draw[fill] (Q) circle (1.5pt);
\coordinate (P) at ($(O)!(Q)!(150:8)$);
\draw[fill,blue] (P) circle (1.5pt);
\draw[name path=path_PQ] (P) -- (Q);

%A right-angle mark is drawn at P.
\coordinate (U) at ($(P)!4mm!45:(O)$);
\draw[dashed] (U) -- ($(P)!(U)!(O)$);
\draw[dashed] (U) -- ($(P)!(U)!(Q)$);

\coordinate (A) at ($(O)!(Q)!(190:4)$);
\coordinate (R) at ($(A)!(P)!(Q)$);
\draw[fill,blue] (R) circle (1.5pt);
\draw (P) -- (R);
\draw (Q) -- (R);
\draw (R) -- (A);

%A right-angle mark is drawn at R.
\coordinate (U_4) at ($(R)!4mm!45:(P)$);
\draw[dashed] (U_4) -- ($(R)!(U_4)!(Q)$);
\draw[dashed] (U_4) -- ($(R)!(U_4)!(P)$);

%The label for O is typeset. An invisible path is drawn 3mm below the line containing
%ray_1. Since the angle for y is between rays inclined at 130 degrees and 150 degrees,
%the label for O is centered along a ray inclined at an angle of -40 degrees.
\coordinate (label_for_O) at ($(0,0)!3mm!-90:(10:1)$);
\node at (label_for_O){$O$};

%The label for Q is typeset.
\coordinate (label_Q_left) at ($(Q)!-1cm!(A)$);
\coordinate (label_Q_right) at ($(Q)!-1cm!(P)$);
\coordinate (label_Q) at ($(label_Q_left)!0.5!(label_Q_right)$);
\node[blue] at ($(Q)!3mm!(label_Q)$){$Q$};

%The label for P is typeset.
\coordinate (label_P_above) at ($(P)!-15mm!(Q)$);
\coordinate (label_P_below) at ($(P)!-15mm!(R)$);
\coordinate (label_P) at ($(label_P_above)!0.5!(label_P_below)$);
\node[blue] at ($(P)!3mm!(label_P)$){$P$};

%The label for R is typeset.
\coordinate (label_R) at ($(R)!-4mm!(P)$);
\node[blue] at (label_R){$R$};

%The angle at O with a measure of 180 - x is drawn. The angle is marked with "|".
\draw[draw=blue] (O) ++(190:0.75) arc (190:150:0.75);
\draw[draw=blue] ($(170:0.75) +(170:-3pt)$) -- ($(170:0.75) +(170:3pt)$);

%The angle at O with a measure of y is drawn.
\draw[draw=blue] (O) ++(150:0.9) arc (150:130:0.9);
\coordinate (label_for_y) at (140:1.1);
\node[font=\footnotesize] at (label_for_y){$y$};

%An angle at P with measure x is drawn. The angle is marked with "|".
\draw[draw=blue] let \p1=($(R)-(P)$), \n1={atan(\y1/\x1)} in ($(P)!0.5cm!(O)$) arc (-30:\n1:0.5);
\draw[draw=blue] let \p1=($(R)-(P)$), \n1={atan(\y1/\x1)} in ($($(P) +({0.5*(\n1-30)}:0.5)$)!-3pt!(P)$) -- ($($(P) +({0.5*(\n1-30)}:0.5)$)!3pt!(P)$);

%An angle at Q with measure x is drawn. The angle is marked with "|".
\draw[draw=blue]
let
\p1=($(Q)-(R)$),
\n1={atan(\y1/\x1)},
\p2=($(Q)-(P)$),
\n2={atan(\y2/\x2)}
in
($(Q)!0.5cm!(R)$) arc (\n1:{\n2 - 180}:0.5);

%\draw[draw=blue,fill]
%  let \p1=($(Q)-(R)$),
%  \n1={atan(\y1/\x1)},
%  \p2=($(Q)-(P)$),
%  \n2={atan(\y2/\x2)}
%  in
%  ($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!-3pt!(Q)$) --
%  ($($(Q) +({0.5*(\n1+\n2+180)}:-0.5)$)!3pt!(Q)$);

\draw[draw=green,fill]
let
\p1=($(Q)-(R)$),
\n1={atan(\y1/\x1)},
\p2=($(Q)-(P)$),
\n2={atan(\y2/\x2)}
in
($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5cm-5pt})$) --
($(Q) + ({0.5*(\n1+\n2+180)}:{-0.5cm+5pt})$);

\end{tikzpicture}

\end{document} 