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Hyperbolic functions

Definitions of hyperbolic functions \( \sinh x=\frac{e^x – e^{-x}}{2} \) \( \cosh x=\frac{e^x + e^{-x}}{2} \) \( \tanh x=\frac{e^x – e^{-x}}{e^x + e^{-x}} =\frac{\sinh x}{\cosh x} \) \( \mathrm{csch}\,x=\frac{2}{e^x – e^{-x}} = \frac{1}{\sinh x} \) \( \mathrm{sech}\,x=\frac{2}{e^x + e^{-x}} = \frac{1}{\cosh x} \) \( \coth\,x=\frac{e^x + e^{-x}}{e^x – e^{-x}} = \frac{\cosh x}{\sinh x} \) Derivatives \( \frac{d}{dx}\, \sinh x = \cosh […]

Trigonometry formulas

Right-Triangle Definitions \( \sin \alpha = \frac{\text{Opposite}}{\text{Hypotenuse}} \) \( \cos \alpha = \frac{\text{Adjacent}}{\text{Hypotenuse}} \) \( \tan \alpha = \frac{\text{Opposite}}{\text{Adjacent}} \) \( \csc \alpha = \frac{1}{\sin\alpha} = \frac{\text{Hypotenuse}}{\text{Opposite}} \) \( \sec \alpha = \frac{1}{\cos\alpha} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \) \( \cot \alpha = \frac{1}{\tan\alpha} = \frac{\text{Adjacent}}{\text{Opposite}} \) Reduction Formulas \( \sin(-x) = -\sin(x) \) \( \cos(-x) = \cos(x) \) \( \sin\left(\frac{\pi}{2} – x\right) = \cos(x) \) \( […]

Logarithm formulas

\( y = \log_a x \Longleftrightarrow a^y = x ~~(a, x >0 , a \ne 1) \) \( \log_a 1 = 0 \) \( \log_a a = 1 \) \( \log_a (mn) = \log_a m + \log_a n \) \( \log_a \frac{m}{n} = \log_a m – \log_a n \) \( \log_a m^n = n \cdot \log_a m \) \( \log_a m […]

Roots Formulas

Notation: a,b : bases \( ( a \geq 0 , b \geq 0 ~~\text{if} ~~ n = 2k ) \) n,m: powers Formulas \( \left( \sqrt[\scriptstyle n]{a} \right)^n = a \) \( \left( \sqrt[\scriptstyle n]{a} \right)^m = \sqrt[\scriptstyle n]{a^m} \) \( \sqrt[\scriptstyle m]{ \sqrt[\scriptstyle n]{a}} = \sqrt[\scriptstyle {n m}]{a} \) \( \left( \sqrt[\scriptstyle n]{a^m} \right)^p = \sqrt[\scriptstyle n]{a^{n p}} \) […]

Exponential Formulas

  \( a^p = \underbrace{a \cdot a \cdot \dots a }_p ~~~( \text{if} ~~ p \in \mathbb{N} ) \)   \( a^0 = 1 ~~~ (\text{if} ~~~ a \ne 0) \)   \( a^r \cdot a^s = a^{r+s} \)   \( \frac{a^r}{a^s} = a^{r-s} \)   \( \left( a^r\right)^s = a^{r \cdot s} \)   \( (a \cdot b)^r = […]