Right-Triangle Definitions :


\[\sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}}=\frac{\text{a}}{\text{c}}\] \[ \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}}=\frac{\text{b}}{\text{c}}\] \[ \tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}}=\frac{\text{a}}{\text{b}}\] \[\csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}}=\frac{\text{c}}{\text{a}}\] \[\sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}}=\frac{\text{c}}{\text{b}}\] \[\cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}}=\frac{\text{b}}{\text{a}}\]

Reduction Formulas :


\[\sin(-x) = -\sin(x)\] \[\cos(-x) = \cos(x)\] \[\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\] \[\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\] \[\sin\left(\frac{\pi}{2} + x\right) = \cos(x)\] \[\cos\left(\frac{\pi}{2} + x\right) = -\sin(x)\] \[\sin(\pi - x) = \sin(x)\] \[\cos(\pi - x) = -\cos(x)\] \[\sin(\pi + x) = -\sin(x)\] \[\cos(\pi + x) = -\cos(x)\]

Basic Identities :


\[ \sin^2x + \cos^2x = 1\] \[ \tan^2x + 1 = \frac{1}{\cos^2x} \] \[ \cot^2x + 1 = \frac{1}{\sin^2x}\]

Sum and Difference Formulas :


\[ \sin(\alpha + \beta) = \sin\alpha \cdot \cos \beta + \sin\beta \cdot \cos\alpha\] \[\sin(\alpha - \beta) = \sin\alpha \cdot \cos \beta - \sin \beta \cdot \cos\alpha\] \[\cos(\alpha + \beta) = \cos\alpha \cdot \cos \beta - \sin\alpha \cdot \cos\beta\] \[ \cos(\alpha - \beta) = \cos\alpha \cdot \cos \beta + \sin\alpha \cdot \cos\beta\] \[ \tan(\alpha + \beta) = \frac{ \tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta }\] \[ \tan(\alpha - \beta) = \frac{ \tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta }\]

Double Angle and Half Angle Formulas :


\[ \sin(2\,\alpha) = 2 \cdot \sin\alpha \cdot \cos\alpha\] \[ \cos(2\,\alpha) = \cos^2\alpha - \sin^2\alpha\] \[ \tan(2\,\alpha) = \frac{2\,\tan\alpha}{1 - \tan^2\alpha}\] \[ \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1-\cos\alpha}{2}}\] \[ \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1+\cos\alpha}{2}}\] \[ \tan \frac{\alpha}{2} = \frac{1 - \cos\alpha}{\sin\alpha} = \frac{\sin\alpha}{1 - \cos\alpha}\] \[ \tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos\alpha}{1 - \cos\alpha} }\]

Other Useful Trig Formulas :


Law of sines \[\frac{\sin\alpha}{\alpha} = \frac{\sin\beta}{\beta} = \frac{\sin\gamma}{\gamma}\]
Law of cosines \[\begin{aligned} a^2 = b^2 + c^2 - 2\cdot b\cdot c\cdot \cos\alpha \\ b^2 = a^2 + c^2 - 2\cdot a\cdot c\cdot \cos\beta \\ c^2 = a^2 + b^2 - 2\cdot a\cdot b\cdot \cos\gamma \end{aligned}\]
Area of triangle \[A = \frac{1}{2} a\,b\, \sin\gamma \]

Quantitative Aptitude
Reasoning
Programming
Interview