<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%This command provides the text to be typeset on the
%first column of page 4. The first trigonometric
%formulae
%
%The macro has one parameter:
%      1) The width of math text
\newcommand\TFourTrigOne[1]{%
   \parbox[t]{#1}{%
      \DisplaySpace{\TFourDisplaySpace}{\TFourDisplayShortSpace}
      %The column is narrow, ragged right looks nicer
      \raggedright

      \input{unit.tex}
      \hspace{-.85em plus .1em minus .1em}\usebox\UnitBox

      \TFourTitle{Pythagorean theorem:}
       \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
          \Fm{C^2 = A^2 + B^2}.
      \end{DisplayFormulae}

      \TFourTitle{Definitions:}
       \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\BigChar}{\StyleWithoutNumber}
         \def\LineOfArray##1##2##3##4{%
              \rule{0pt}{4ex plus 1ex minus .5ex}%
              {##1}&amp;=&amp;{##2}&amp;{##3}&amp;=&amp;{##4}\\}
         \Fm{\begin{array}{l%Equal sign
                           @{\hspace{.1em}}c@{\hspace{.2em}}%
                           l%
                           l%Equal sign
                           @{\hspace{.1em}}c@{\hspace{.2em}}%
                           l}
                  \LineOfArray{\sin a}{\frac{A}{C}}{\cos a}{\frac{B}{C}}
                  \LineOfArray{\csc a}{\frac{C}{A}}{\sec a}{\frac{C}{B}}
                  \LineOfArray{\tan a}{\frac{\sin a}{\cos a} = \frac{A}{B}}
                               {\cot a}{\frac{\cos a}{\sin a}=\frac{B}{A}} 
             \end{array}%
            }
      \end{DisplayFormulae}

       \TFourTitle{Area, radius of inscribed circle:}
        \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
           \Fm{\tfrac{1}{2} A B} 
           \Fm{\frac{A B}{A + B + C}}
        \end{DisplayFormulae}

      \TFourTitle{Identities:}
      \begin{DisplayFormulae}{1}{0pt}{\TFourSkipFormulae}{\BigChar}{\StyleWithoutNumber}
         \def\FmSep{\text{,}}
         \Fm{\sin x = \frac{1}{\csc x}}
         \Fm{\cos x = \frac{1}{\sec x}}
         \Fm{\tan x = \frac{1}{\cot x}}
         \Fm{\sin^2 x + \cos^2 x = 1}
         \Fm{1 + \tan^2 x = \sec^2 x}
         \Fm{1 + \cot^2 x = \csc^2 x}
         \Fm{\sin x = \cos \left(\tfrac{\pi}{2} - x\right)}
         \Fm{\sin x = \sin (\pi - x)}
         \Fm{\cos x = - \cos (\pi - x)}
         \Fm{\tan x = \cot \left(\tfrac{\pi}{2} - x\right)}
         \Fm{\cot x = - \cot (\pi - x)}
         \Fm{\csc x = \cot \tfrac{x}{2} - \cot x}
         \Fm{\sin (x \pm y) = \sin x \cos y \pm \cos x \sin y}
         \Fm{\cos (x \pm y) = \cos x \cos y \mp \sin x \sin y}
         \Fm{\tan (x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x \tan y}}
         \Fm{\sin 2 x =  2 \sin x \cos x}
         \Fm{\sin 2 x = \frac{2 \tan x}{1 + \tan^2 x}}
         \Fm{\cot (x \pm y) = \frac{\cot x \cot y \mp 1}{\cot x \pm \cot y }}
         \Fm{\cos 2 x = \cos^2 x  - \sin^2 x}
         \Fm{\cos 2 x = 2 \cos^2 x - 1}
         \Fm{\cos 2 x = 1 - 2 \sin^2 x}
         \Fm{\cos 2 x = \frac{1 -  \tan^2 x}{1 + \tan^2 x}}
         \Fm{\tan 2 x = \frac{2 \tan x}{1 - \tan^2 x}}
         \Fm{\cot 2 x = \frac{\cot^2 x - 1}{2 \cot x}}
         \Fm{\sin (x + y) \sin (x - y) = \sin^2 x - \sin^2 y}
         \def\FmSep{\text{.}}
         \Fm{\cos (x + y) \cos (x - y) = \cos^2 x - \sin^2 y}
      \end{DisplayFormulae}

       \TFourTitle{Euler's equation:}
        \begin{DisplayFormulae}{1}{0pt}{3ex plus 1ex minus .5ex}{\SmallChar}{\StyleWithoutNumber}
            \Fm{e^{i x} = \cos x + i \sin x},  
            \Fm{e^{i \pi} + 1= 0}.
       \end{DisplayFormulae}
   }
}
</pre></body></html>