Essential Math === ## [Trigonometry](https://en.wikipedia.org/wiki/Trigonometry) ### Right Triangle  $\sin \theta = \displaystyle\frac{\textsf{opposite}}{\textsf{hypotenuse}}$ $\csc \theta = \displaystyle\frac{\textsf{hypotenuse}}{\textsf{opposite}}$ $\cos \theta = \displaystyle\frac{\textsf{adjacent}}{\textsf{hypotenuse}}$ $\sec \theta = \displaystyle\frac{\textsf{hypotenuse}}{\textsf{adjacent}}$ $\tan \theta = \displaystyle\frac{\textsf{opposite}}{\textsf{adjacent}}$ $\cot \theta = \displaystyle\frac{\textsf{adjacent}}{\textsf{opposite}}$ ### Each Trigonometric Function in Terms of the other Five  ### [Unit Circle](https://en.wikipedia.org/wiki/Unit_circle)   ### [Trigometric Identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities) #### Pythagorean Identities $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \cot^2 \theta = \csc^2 \theta$ $\tan^2 \theta + 1 = \sec^2 \theta$ #### Half-Angle Identities $\sin^2 \theta = \displaystyle\frac{1 - \cos 2\theta}{2}$ $\cos^2 \theta = \displaystyle\frac{1 + \cos 2\theta}{2}$ $\sin \theta \cos \theta = \displaystyle\frac{\sin 2\theta}{2}$ ## [Logarithms](https://www.rapidtables.com/math/algebra/Logarithm.html#log-rules) ### [Logarithm Definition](https://www.rapidtables.com/math/algebra/Logarithm.html#log-definition) The logarithm is the inverse of the exponential function. $$ \textsf{If} \hspace{3mm} b^y = x \hspace{3mm} \textsf{then}\hspace{3mm} \log_b(x) = y $$ [Euler's Number](https://en.wikipedia.org/wiki/E_(mathematical_constant)), $e$, is often used as the base. $$ \ln{x} = \log_e{x} \hspace{3mm} \textsf{where} \hspace{3mm} e=2.71828... $$ When no base is specified, ten is typically assumed. This is not true in [Julia code](https://julialang.org/), where the [log function](https://docs.julialang.org/en/v1/manual/mathematical-operations/#Powers,-logs-and-roots) implies base $e$. $$ \log{x} = \log_{10}{x} $$ ### [Logarithm product rule](https://www.rapidtables.com/math/algebra/Logarithm.html#product-rule) $\log_{b}(xy) = \log_{b}(x) + \log_{b}(y)$ ### [Logarithm quotient rule](https://www.rapidtables.com/math/algebra/Logarithm.html#quotient-rule) $\log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y)$ ### [Logarithm power rule](https://www.rapidtables.com/math/algebra/Logarithm.html#power-rule) $\log_{b}(x^y) = y\log_{b}(x)$ ### [Logarithm base switch rule](https://www.rapidtables.com/math/algebra/Logarithm.html#base-switch) $\log_{b}(c) = \displaystyle\frac{1}{\log_{c}(b)}$ ### [Logarithm base change rule](https://www.rapidtables.com/math/algebra/logarithm/Logarithm_Base_Change.html) $\log_{b}(x) = \displaystyle\frac{\log_{c}(x)}{\log_{c}(b)}$ ### [Logarithm of One](https://www.rapidtables.com/math/algebra/Logarithm.html#log-1) $\log_{b}(1) = 0$ ### [Logarithm of Zero](https://www.rapidtables.com/math/algebra/Logarithm.html#log-0) $\large\lim_{x\to 0^{+}} \hspace{2mm} \normalsize \log_{b}(x) = - \infty$ ### [Logarithm of Negative Values](https://www.rapidtables.com/math/algebra/Logarithm.html#log-negative) $\log_{b}(x) \textsf{ is undefined when }x \leq 0$