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% Last Updated 2015 Jan 01

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\chead{Trigonometry Formulas by www.ReeTutorsMath.org}
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\begin{document}
\centering

\subsection*{\quad \qquad Definitions \newline (OPPosite, ADJacent, HYPotenouse)}
\begin{equation*}
\begin{array}{c c c}
\sin(\theta) = \displaystyle \frac{\text{opp}}{\text{hyp}} & \qquad
  & \csc(\theta) = \displaystyle \frac{\text{hyp}}{\text{opp}}\\
\noalign{\medskip}
\cos(\theta) = \displaystyle \frac{\text{adj}}{\text{hyp}} & \qquad
  & \sec(\theta) = \displaystyle \frac{\text{hyp}}{\text{adj}}\\
\noalign{\medskip}
\tan(\theta) = \displaystyle \frac{\text{opp}}{\text{adj}} & \qquad
  & \cot(\theta) = \displaystyle \frac{\text{adj}}{\text{opp}}\\
  \\
\end{array}
\end{equation*}

\subsection*{Definitions - (x, y, r)}
\begin{equation*}
\begin{array}{c c c}
\sin(\theta) = \displaystyle \frac{y}{r} & \qquad
  & \csc(\theta) = \displaystyle \frac{r}{y}\\
\noalign{\medskip}
\cos(\theta) = \displaystyle \frac{x}{r} & \qquad
  & \sec(\theta) = \displaystyle \frac{r}{x}\\
\noalign{\medskip}
\tan(\theta) = \displaystyle \frac{y}{x} & \qquad
  & \cot(\theta) = \displaystyle \frac{x}{y}\\
  \\
\end{array}
\end{equation*}

\subsection*{Changing To Sine And Cosine}
\begin{equation*}
\begin{array}{c c c}
\tan(\theta) = \displaystyle \frac{\sin(\theta)}{\cos(\theta)} & \qquad
  & \sec(\theta)= \displaystyle \frac{1}{\cos(\theta)}\\
\\
\cot(\theta)= \displaystyle \frac{\cos(\theta)}{\sin(\theta)} & \qquad
  & \csc(\theta) = \displaystyle \frac{1}{\sin(\theta)}\\
  \\
\end{array}
\end{equation*}

\subsection*{Pythagorean Identities}
\begin{equation*}
\begin{array}{r c l}
\sin^2(\theta) +  \cos^2(\theta) & = &1 \\
\noalign{\medskip}
\tan^2(\theta) + 1               & = & \sec^2(\theta)\\
\noalign{\medskip}
1              + \cot^2(\theta)  &=& \csc^2(\theta) \\
\noalign{\medskip}
\sin(\theta) &=& ^+_- \sqrt{1-\cos^2(\theta)}\\
\noalign{\medskip}
\cos(\theta) &=& ^+_- \sqrt{1-\sin^2(\theta)}\\
\\
\end{array}
\end{equation*}

\subsection*{Double-Angle Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin(2\theta)&=& 2\sin(\theta)\cos(\theta)\\
\noalign{\medskip}
\cos(2\theta)&=& \cos^2(\theta)-\sin^2(\theta)\\
\noalign{\medskip}
\cos(2\theta)&=& 2\cos^2(\theta)-1\\
\noalign{\medskip}
\cos(2\theta)&=& 1-2\sin^2(\theta)\\
\noalign{\medskip}
\tan(2\theta)&=& \displaystyle \frac{2\tan(\theta)}{1-\tan^2(\theta)}\\
\end{array}
\end{equation*}

\subsection*{Half-Angle Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin\left( \displaystyle \frac{\theta}{2}\right) &=& ^+_- \sqrt{ \displaystyle \frac{1-\cos(\theta)}{2}}\\
\noalign{\medskip}
\cos\left( \displaystyle \frac{\theta}{2}\right) &=& ^+_- \sqrt{ \displaystyle \frac{1+\cos(\theta)}{2}}\\
\noalign{\medskip}
\tan\left( \displaystyle \frac{\theta}{2}\right) &=& \displaystyle \frac{1-\cos(\theta)}{\sin(\theta)}\\
\noalign{\medskip}
\tan\left( \displaystyle \frac{\theta}{2}\right) &=& \displaystyle \frac{\sin(\theta)}{1+\cos(\theta)}\\
\end{array}
\end{equation*}

\subsection*{Sum and Difference Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin(A+B) &=& \sin(A)\cos(B) + \cos(A)\sin(B)\\
\noalign{\medskip}
\sin(A-B) &=& \sin(A)\cos(B) - \cos(A)\sin(B)\\
\noalign{\medskip}
\cos(A+B) &=& \cos(A)\cos(B) - \sin(A)\sin(B)\\
\noalign{\medskip}
\cos(A-B) &=& \cos(A)\cos(B) + \sin(A)\sin(B)\\
\\
\tan(A+B) &=& \displaystyle \frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}\\
\\
\tan(A-B) &=& \displaystyle \frac{\tan(A)-\tan(B)}{1+\tan(A)\tan(B)}\\
\\
\end{array}
\end{equation*}

\subsection*{Product-to-Sum Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin(A)\sin(B) &=& \displaystyle \frac{1}{2} [\cos(A-B)-\cos(A+B)]\\
\\
\cos(A)\cos(B) &=& \displaystyle\frac{1}{2} [\cos(A-B)+\cos(A+B)]\\
\\
\sin(A)\cos(B) &=& \displaystyle\frac{1}{2} [\sin(A-B)+\sin(A+B)]\\
\end{array}
\end{equation*}

\subsection*{Sum-to-Product Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin(A)+\sin(B) &=& 2\sin\left(\displaystyle \frac{A+B}{2}\right)\cos\left(\displaystyle \frac{A-B}{2}\right)\\
\noalign{\medskip}
\sin(A)-\sin(B) &=& 2\cos\left(\displaystyle \frac{A+B}{2}\right)\sin\left(\displaystyle \frac{A-B}{2}\right)\\
\noalign{\medskip}
\cos(A)+\cos(B) &=& 2\cos\left(\displaystyle \frac{A+B}{2}\right)\cos\left(\displaystyle \frac{A-B}{2}\right)\\
\noalign{\medskip}
\cos(A)-\cos(B) &=& - 2\sin\left(\displaystyle \frac{A+B}{2}\right)\sin\left(\displaystyle \frac{A-B}{2}\right)\\
\noalign{\medskip}
\end{array}
\end{equation*}

\subsection*{Power Reducing Formulas}
\begin{equation*}
\begin{array}{r c l}
\sin^2(\theta) & =& \displaystyle \frac{1-\cos(2\theta)}{2}\\
\noalign{\medskip}
\cos^2(\theta) & =& \displaystyle \frac{1+\cos(2\theta)}{2}\\
\noalign{\medskip}
\tan^2(\theta) & =& \displaystyle \frac{1-\cos(2\theta)}{1+\cos(2\theta)}\\
\noalign{\medskip}
\end{array}
\end{equation*}

\subsection*{Even and Odd Trig Functions}
\begin{equation*}
\begin{array}{r c l r}
\sin(-\theta) &=&\ -\sin(\theta) & \text{sine is ODD}\\
\csc(-\theta) &=& -\csc(\theta)& \text{cosecant is ODD}\\
\tan(-\theta) &=& -\tan(\theta)&\text{tangent is ODD}\\
\cot(-\theta) &=& -\cot ( \theta )& \qquad \text{cotangent is ODD}\\
\cos(-\theta) &=& \cos(\theta) & \text{cosine is EVEN}\\
\sec(-\theta) &=& \sec(\theta) & \text{secant is EVEN}\\
\end{array}
\end{equation*}

\subsection*{Cofunction formulas(in degrees)}
\begin{equation*}
\begin{array}{r c l}
\sin(\theta) & = & \cos(90^{\circ}-\theta)\\
\cos(\theta) & = & \sin(90^{\circ}-\theta)\\
\tan(\theta) & = & \cot(90^{\circ}-\theta)\\
\cot(\theta) & = & \tan(90^{\circ}-\theta)\\
\sec(\theta) & = & \csc(90^{\circ}-\theta)\\
\csc(\theta) & = & \sec(90^{\circ}-\theta)\\
\end{array}
\end{equation*}

\subsection*{Cofunction formulas(in radians)}
\begin{equation*}
\begin{array}{r c l}
\sin(\theta) & = & \cos\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\cos(\theta) & = & \sin\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\tan(\theta) & = & \cot\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\cot(\theta) & = & \tan\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\sec(\theta) & = & \csc\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\csc(\theta) & = & \sec\left(\displaystyle\frac{\pi}{2}-\theta\right)\\ \\
\end{array}
\end{equation*}

\subsection*{Derivative formulas (Calculus)}
\begin{equation*}
\begin{array}{r c l}
\displaystyle \frac{d}{dx} \Big(\sin(x)\Big) & = & \cos(x)\\ \\
\displaystyle \frac{d}{dx} \Big(\cos(x)\Big) & = & -\sin(x)\\ \\
\displaystyle \frac{d}{dx} \Big(\tan(x)\Big) & = & \sec^2(x)\\ \\
\displaystyle \frac{d}{dx} \Big(\cot(x)\Big) & = & -csc^2(x)\\ \\
\displaystyle \frac{d}{dx} \Big(\sec(x)\Big) & = & \sec(x) \tan(x)\\ \\
\displaystyle \frac{d}{dx} \Big(\csc(x)\Big) & = & -\csc(x) \cot(x)\\ \\
\end{array}
\end{equation*}


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