Trigonometric Functions, Identities & Equations

Special Angles

$$\begin{array}{|c|c|c|c|c|c|} \hline \theta & 0^{\circ}& 30^{\circ} & 45^{\circ} & 60^{\circ} & 90^{\circ} \\ \hline \sin\theta & \frac{\sqrt{0}}{2}=0 & \frac{\sqrt{1}}{2}=\frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & \frac{\sqrt{4}}{2}=1 \\ \hline \cos\theta & \frac{\sqrt{4}}{2}=1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{1}}{2}=\frac{1}{2} & \frac{\sqrt{0}}{2}=0 \\ \hline \tan\theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & - \\ \hline \end{array}$$

Reciprocal Functions

$\,\text{cosec }\theta=\dfrac{1}{\sin\theta}$

$\,\sec\theta=\dfrac{1}{\cos\theta}$

$\,\cot\theta=\dfrac{1}{\tan\theta}$

Negative Functions

$\,\sin(-\theta)=-\sin\theta$

$\,\cos(-\theta)=\cos\theta$

$\,\tan(-\theta)=-\tan\theta$

Tangent & Cotangent

$\,\tan\theta=\dfrac{\sin\theta}{\cos\theta}$

$\,\cot\theta=\dfrac{\cos\theta}{\sin\theta}$

Trigonometric Identities

$\,\sin^2{\theta} + \cos^2{\theta} = 1$

$\,\sec^2{\theta}=1+\tan^2{\theta}$

$\,\text{cosec}^2{\theta}=1 + \cot^2{\theta}$

Addition Formulae

$\,\sin(A \pm B) = \sin A\cos B \pm \cos A \sin B$

$\,\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\,\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle Formulae

$\,\sin 2A = 2\sin A \cos A$

$\,\cos2A = \cos^2A - \sin^2A = 2\cos^2A - 1 = 1 - 2\sin^2A$

$\,\tan 2A = \dfrac{2\tan A}{1-\tan^2A}$

R-Formulae

$\,a\sin \theta \pm b \cos \theta = R \sin (\theta \pm \alpha)$

$\,a \cos \theta \pm b \sin \theta = R \cos (\theta \mp \alpha)$

where $R=\sqrt{a^2+b^2}, \tan\alpha=\dfrac{b}{a}$.