Trigonometric Formulae

Addition Formulae

\begin{align} cos(A+B) & = cosAcosB-sinAsinB \\ \\ cos(A-B) & = cosAcosB +sinAsinB \\ \\ sin(A+B) & = sinAcosB +sinBcosA \\ \\ sin(A-B) & = sinAcosB-sinBcosA \\ \\ tan(A+B) & = \frac{tanA+tanB}{1-tanAtanB} \\ \\ tan(A-B) & = \frac{tanA -tanB}{1+tanAtanB} \\ \end{align}

Half-angle Formulae

\begin{align} tan(\theta) & = \frac{tan \frac{\theta}{2}+tan \frac{\theta}{2}}{1-tan \frac{\theta}{2} tan \frac{\theta}{2}} \\ \\ sin(\theta) & =\frac{tan \frac{\theta}{2}+tan \frac{\theta}{2}}{1+tan \frac{\theta}{2} tan \frac{\theta}{2}} \\ \\ cos(\theta) & = \frac {1-tan \frac{\theta}{2} tan \frac{\theta}{2}} {1+tan \frac{\theta}{2} tan \frac{\theta}{2}} \\ \end{align}

Sum Formulae

\begin{align} sin(x) + sin(y) = 2 sin\ \left(\frac{x + y}{2} \right) \cos\ \left(\frac{x - y}{2} \right) \\ \\ sin(x) - sin(y) = 2 cos\ \left(\frac{x + y}{2} \right) sin\ \left(\frac{x - y}{2} \right) \\ \\ cos(x) + cos(y) = 2 cos\ \left(\frac{x + y}{2}\ \right) cos \left(\frac{x - y}{2} \right) \\ \\ cos(x) - cos(y) = -2 sin\ \left(\frac{x + y}{2}\ \right)\ sin \left(\frac{x - y}{2}\ \right) \end{align}

Product Formulae

\begin{align} sin(x)\ cos(y) = \frac{1}{2} \left[sin(x + y) + sin(x - y) \right] \\ \\ cos(x)\ sin(y) = \frac{1}{2} \left[sin(x + y) - sin(x - y) \right] \\ \\ cos(x)\ cos(y) = \frac{1}{2} \left[cos(x - y) + cos(x + y) \right] \\ \\ sin(x)\ sin(y) = \frac{1}{2} \left[cos(x - y) - cos(x + y) \right] \\ \end{align}