Trigonometric Function

\sin (θ) = \cos (\frac{π}{2} - θ) = \frac{1}{\csc (θ)} .

\cos (θ) = \sin (\frac{π}{2} - θ) = \frac{1}{sec (θ)} .

\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = cot (\frac{π}{2} - θ) = \frac{1}{cot (θ)} .

\sin^{2} (θ) + \cos^{2} (θ) = 1 .

\sin (θ) = \sin (θ + 2 k π),

\cos (θ) = \cos (θ + 2 k π),

\tan (θ) = \tan (θ + k π) .

Values

\begin{matrix} θ & \sin (θ) & \cos (θ) & \tan (θ) \\ 0 & 0 & 1 & 0 \\ \frac{π}{6} & \frac{1}{2} & \frac{\sqrt[]{3}}{2} & \frac{1}{\sqrt[]{3}} \\ \frac{π}{4} & \frac{1}{\sqrt[]{2}} & \frac{1}{\sqrt[]{2}} & 1 \\ \frac{π}{3} & \frac{\sqrt[]{3}}{2} & \frac{1}{2} & \sqrt[]{3} \\ \frac{π}{2} & 1 & 0 & - \end{matrix}

\frac{d}{dx} \sin (x) = \cos (x),

\frac{d}{dx} \cos (x) = - \sin (x),

\sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots = Σ_{n = 0}^{∞} \frac{{(- 1)}^{n}}{(2 n + 1)!} x^{2 n + 1} .

\cos (x) = 1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!} + \dots = Σ_{n = 0}^{∞} \frac{{(- 1)}^{n}}{(2 n)!} x^{2 n} .

\sin (z) = z Π_{n = 1}^{∞} (1 - \frac{z^{2}}{n^{2} π^{2}}), z : ℂ,

\cos (z) = Π_{n = 1}^{∞} (1 - \frac{z^{2}}{{(n - 1 / 2)}^{2} π^{2}}), z : ℂ,

e^{i x} = \cos (x) + i \sin (x) .

\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y) .

\begin{matrix} \sin (- x) = - \sin (x), \\ \cos (- x) = \cos (x), \\ \tan (- x) = - \tan (x) . \end{matrix}

\begin{matrix} \sin (x + y) = \sin (x) \cos (y) + \cos (x) \sin (y), \\ \cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y), \\ \tan (x + y) = \frac{\tan (x) + \tan (y)}{1 - \tan (x) \tan (y)} . \end{matrix}

\begin{matrix} \sin (x - y) = \sin (x) \cos (y) - \cos (x) \sin (y), \\ \cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y), \\ \tan (x - y) = \frac{\tan (x) - \tan (y)}{1 + \tan (x) \tan (y)} . \end{matrix}

\begin{matrix} \sin (2 x) = \sin (x) \cos (x) + \cos (x) \sin (x) = 2 \sin (x) \cos (x) =, \\ \cos (2 x) = \cos (x) \cos (x) - \sin (x) \sin (x) = \cos^{2} (x) - \sin^{2} (x) =, \\ \tan (2 x) = \frac{\tan (x) + \tan (x)}{1 - \tan (x) \tan (x)} = \frac{2 \tan (x)}{1 - \tan^{2} (x)} . \end{matrix}

\begin{matrix} \sin (t) = \frac{2 t}{1 + t^{2}}, \\ \cos (t) = \frac{1 - t^{2}}{1 + t^{2}}, \\ \tan (t) = \frac{2 t}{1 - t^{2}} . \end{matrix}