Multiple angle formulas

Definition

A trigonometric identity which appears in terms of multiple angle is called a multiple angle identity. A multiple angle identity is also called as multiple angle formula.

List of identities

There are several types of multiple angle formulae in trigonometry but double angle and triple angle trigonometric rules are mostly and mainly useful in mathematics. The remaining multiple angle identities can be derived by using these two types of multiple angle laws.

01

Double angle formulas

$(1)\,\,\,\,$ $\sin{2\theta}$ $\,=\,$ $2\sin{\theta}\cos{\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1+\tan^2{\theta}}$

$(2)\,\,\,\,$ $\cos{2\theta}$ $\,=\,$ $\cos^2{\theta}-\sin^2{\theta}$ $\,=\,$ $2\cos^2{\theta}-1 = 1 -2\sin^2{\theta}$

$(3)\,\,\,\,$ $\tan{2\theta}$ $\,=\,$ $\dfrac{2\tan{\theta}}{1-\tan^2{\theta}}$ $\,=\,$ $\dfrac{2}{\cot{\theta}-\tan{\theta}}$

$(4)\,\,\,\,$ $\cot{2\theta}$ $\,=\,$ $\dfrac{\cot^2{\theta}-1}{2\cot{\theta}}$ $\,=\,$ $\dfrac{\cot{\theta}-\tan{\theta}}{2}$

02

Triple angle formulas

$(1)\,\,\,\,$ $\sin{3\theta}$ $\,=\,$ $3\sin{\theta}-4\sin^3{\theta}$

$(2)\,\,\,\,$ $\cos{3\theta}$ $\,=\,$ $4\cos^3{\theta}-3\cos{\theta}$

$(3)\,\,\,\,$ $\tan{3\theta}$ $\,=\,$ $\dfrac{3\tan{\theta}-\tan^3{\theta}}{1-3\tan^2{\theta}}$

$(4)\,\,\,\,$ $\cot{3\theta}$ $\,=\,$ $\dfrac{3\cot{\theta}-\cot^3{\theta}}{1-3\cot^2{\theta}}$