topics

Trigonometric Identities

Definitions

\begin{array}{l} \frac{1}{\cos} = \sec \\ \frac{1}{\sin} = cosec \\ \frac{1}{\tan} = cotan \end{array}

\begin{array}{l} \cos (- x) = \cos x \\ \sin (- x) = - \sin x \\ \tan (- x) = - \tan x \\ (\cos)' = - \sin \\ (\sin)' = \cos \\ (\tan)' = \sec^{2} \\ \cos^{2} x + \sin^{2} x = 1 \\ 1 + \tan^{2} x = \sec^{2} x \\ \sin (x + y) = \sin x \cos y + \cos x \sin y \\ \sin (2 x) = 2 \sin x \cos x \\ \cos (x + y) = \cos x \cos y - \sin x \sin y \\ \cos (2 x) = \cos^{2} x - \sin^{2} x \\ = 2 \cos^{2} x - 1 = 1 - 2 \sin^{2} x \\ \tan (x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \\ \tan (2 x) = \frac{2 \tan x}{1 - \tan^{2} x} \end{array}

\begin{array}{l} e^{i θ} = \cos θ + i \sin θ \\ e^{i θ} + e^{- i θ} = \cos θ + i \sin θ + \frac{1}{\cos θ + i \sin θ} \\ = \cos θ + i \sin θ + \frac{\cos θ - i \sin θ}{\cos^{2} θ + \sin^{2} θ} \\ = 2 \cos θ \\ e^{i θ} - e^{- i θ} = 2 i \sin θ \end{array}