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% vim: ts=2 sw=2 et :
\documentclass[tikz, border=2mm]{standalone}
\usepackage{amsmath}
\usepackage{bm}
\usepackage{times}
\usepackage{txfonts}
\usepackage{tikz-3dplot}
\tdplotsetmaincoords{60}{130}
\pgfmathsetmacro{\l}{2}
\begin{document}
\begin{tikzpicture}[
>=latex,
tdplot_main_coords,
]
% origin and poles
\coordinate (O) at (0,0,0);
\coordinate (NP) at (0,0,\l);
\coordinate (SP) at (0,0,-\l);
% gray unit circle
\tdplotdrawarc[lightgray, dashed]{(O)}{\l}{0}{360}{}{};
\draw[lightgray, dashed] (-\l, 0, 0) to (\l, 0, 0);
\draw[lightgray, dashed] (0, -\l, 0) to (0, \l, 0);
% axis
\draw[->] (O) -- ++(0,0,1.25*\l) node[above] {\(\mathbf{\hat{z}}\)};
% meridians
\foreach \phi in {0, 30, 60, ..., 150}{
\tdplotsetrotatedcoords{\phi}{90}{0};
\tdplotdrawarc[lightgray, densely dotted, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
}
% dot above and its projection
\pgfmathsetmacro{\phi}{120}
\pgfmathsetmacro{\theta}{40}
\pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
\pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
\pgfmathsetmacro{\pz}{cos(\theta)*\l})
% Special meridian
\tdplotsetrotatedcoords{\phi-90}{90}{0};
\tdplotdrawarc[gray, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
% point A
\coordinate (A) at (\px,\py,\pz);
\coordinate (Ap) at (\px,\py, 0);
% lines
\draw[red!80!black, thick, ->] (O) -- (A);
\draw[red!80!black]
(O) -- node[midway, below, font=\small, sloped, fill=white] {$\sqrt{1 - z^2}$}
(Ap) -- node[midway, right, font=\small, sloped, fill=white, anchor=north] {$z = \cos \vartheta$}
(A) node[above right, fill=white] {$r = 1$};
% theta arc
\tdplotsetrotatedcoords{\phi-90}{-90}{0};
\tdplotdrawarc[blue!80!black, ->, tdplot_rotated_coords]{(O)}{.95\l}{0}{\theta}{}{};
\node[above right = 1mm, blue!80!black] at (0,0,.8\l) {\(\bm{\hat{\vartheta}}\)};
\end{tikzpicture}
\end{document}
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