Double-angle formulas can be expanded to multiple-angle functions (triple, quadruple, quintuple, and so on) by using the angle sum formulas, and then reapplying the double-angle formulas.

\begin{array}{l} s i n (A + B) = s i n A c o s B + c o s A s i n B \end{array}

\begin{array}{l} s i n (A - B) = s i n A c o s B - c o s A s i n B \end{array}

\begin{array}{l} c o s (A + B) = c o s A c o s B - s i n A s i n B \end{array}

\begin{array}{l} c o s (A - B) = c o s A c o s B + s i n A s i n B \end{array}

\begin{array}{l} s i n α + s i n β = 2 s i n \frac{α + β}{2} c o s \frac{α - β}{2} \end{array}

\begin{array}{l} s i n α - s i n β = 2 s i n \frac{α - β}{2} c o s \frac{α + β}{2} \end{array}

\begin{array}{l} c o s α + c o s β = 2 c o s \frac{α + β}{2} c o s \frac{α - β}{2} \end{array}

\begin{array}{l} c o s α - c o s β = - 2 s i n \frac{α + β}{2} s i n \frac{α - β}{2} \end{array}

\begin{array}{l} s i n 2 α = 2 s i n α c o s α \end{array}

\begin{array}{l} c o s 2 α = c o s^{2} α - s i n^{2} α = 2 c o s^{2} α - 1 = 1 - 2 s i n^{2} α \end{array}

\begin{array}{l} t a n 2 α = \frac{2 t a n α}{1 - t a n^{2} α} \end{array}

Half Angle Formulas

\begin{array}{l} s i n (\frac{a}{2}) = \pm \sqrt{\frac{(1 - c o s a)}{2}} \end{array}

\begin{array}{l} c o s (\frac{a}{2}) = \pm \sqrt{\frac{(1 + c o s a)}{2}} \end{array}

\begin{array}{l} t a n (\frac{a}{2}) = \frac{1 - c o s a}{s i n a} = \frac{s i n a}{1 + c o s a} \end{array}

These formulas can also be written as:

\begin{array}{l} s i n (\frac{a}{2}) = \sqrt{\frac{1 - c o s (a)}{2}} \end{array}

\begin{array}{l} c o s (\frac{a}{2}) = \sqrt{\frac{1 + c o s (a)}{2}} \end{array}

\begin{array}{l} t a n (\frac{a}{2}) = \sqrt{\frac{1 - c o s (a)}{1 + c o s (a)}} \end{array}

Half Angle Formulas

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