Function identities - Essential Mathematics for Chemical Engineers

Trigonmetric functions¶

\begin{align} \sin \theta &= \frac{y}{r} & \csc \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\ \cos \theta &= \frac{x}{r} & \sec \theta &= \frac{r}{x} = \frac{1}{\cos \theta} \\ \tan \theta &= \frac{y}{x} = \frac{\sin \theta}{\cos \theta} & \cot \theta &= \frac{x}{y} = \frac{\cos \theta}{\sin \theta} \end{align}

\begin{align} \sin^2 \theta + \cos^2 \theta = 1 \\ 1 + \tan^2 \theta = \sec^2 \theta \\ 1 + \cot^2 \theta = \csc^2 \theta \end{align}

\begin{align} \cos(A+B) = \cos A \cos B - \sin A \sin B \\ \sin(A+B) = \sin A \cos B - \cos A \sin B \end{align}

\begin{align} \cos 2 \theta = \cos^2 \theta - \sin^2 \theta \\ \sin 2 \theta = 2 \sin \theta \cos \theta \end{align}

\begin{align} \cos^2 \theta = \frac{1 + \cos 2 \theta}{2} \\ \sin^2 \theta = \frac{1 - \cos 2 \theta}{2} \end{align}

\begin{align} \sin(\theta + 2 \pi) = \sin \theta \\ \cos(\theta + 2 \pi) = \cos \theta \end{align}

\begin{align} \sin(- \theta) = - \sin \theta \\ \cos(- \theta) = cos \theta \end{align}

Exponential functions¶

\begin{align} a^x a^y &= a^{x+y} \\ \frac{a^x}{a^y} &= a^{x-y} \\ (a^x)^y = (a^y)^x &= a^{xy} \\ a^x b^x &= (ab)^x \\ \frac{a^x}{b^x} &= \left (\frac{a}{b} \right)^x \end{align}

Logarithmic functions¶

Definition:

\begin{align} y &= \log_{a}x \\ x &= a^y \end{align}

Natural Log:

\ln x = \log_{e} x

Common log:

\log x = \log_{10} x

\begin{align} \ln(bx) &= \ln(b) + \ln(x) \\ \ln\left(\frac{b}{x}\right) &= \ln(b) - \ln(x) \\ \ln(x^r) &= r \ln(x) \\ \log_{a}x &= \frac{\ln(x)}{\ln(a)} \\ \end{align}