Problem source

Solve the inequality
$$\frac{\sin\theta+1}{\cos\theta}\le1\;$$
where $0\le\theta<2\pi\,$ and $\cos\theta\ne0\,$.

Solution source

\begin{center}
    \begin{tikzpicture}
    \def\functionf(#1){sin((#1)*180/pi)+1};
    \def\functiong(#1){cos((#1)*180/pi)};
    \def\functionh(#1){(sin((#1)*180/pi)+1)/cos((#1)*180/pi)};
    \def\xl{-3.14/2};
    \def\xu{5*3.14/2};
    \def\yl{-4};
    \def\yu{4};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, blue, smooth, domain=0:{2*pi}, samples=100] 
            plot (\x, {\functionf(\x)});
        \draw[thick, green, smooth, domain=0:{2*pi}, samples=100] 
            plot (\x, {\functiong(\x)});
        \draw[thick, red, smooth, domain=0:{pi/2-0.01}, samples=100] 
            plot (\x, {\functionh(\x)});
        \draw[thick, red, smooth, domain={pi/2+0.01}:{3*pi/2-0.01}, samples=100] 
            plot (\x, {\functionh(\x)});
        \draw[thick, red, smooth, domain={3*pi/2+0.01}:{4*pi/2}, samples=100] 
            plot (\x, {\functionh(\x)});
        \draw[thick, red, dashed] ({pi/2}, \yl) -- ({pi/2}, \yu);
        \draw[thick, red, dashed] ({3*pi/2}, \yl) -- ({3*pi/2}, \yu);
        \draw[thick, black, dashed] (\xl, 1) -- (\xu,1);
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
    
    \end{tikzpicture}
\end{center}

$\theta \in \{0\} \cup (\frac{\pi}{2}, 2\pi]$. In $(0, \frac{\pi}{2})$ $\cos \theta < 1  + \sin \theta$, and then it's either negative or greater than $1+ \sin \theta$