Inline and display LaTeX math to PDF

# A formula sheet

Inline math sits on the baseline of the prose: the quadratic formula gives the roots of $ax^2 + bx + c = 0$ as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, and a right triangle obeys $a^2 + b^2 = c^2$.

## Algebra

$ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k} \qquad \binom{n}{k} = \frac{n!}{k!\,(n-k)!} $

$ \log_b(xy) = \log_b x + \log_b y \qquad a^{m} \cdot a^{n} = a^{m+n} \qquad \sqrt[n]{a^m} = a^{m/n} $

## Trigonometry

$ \sin^2\theta + \cos^2\theta = 1 \qquad \tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta $

The law of cosines, $c^2 = a^2 + b^2 - 2ab\cos\gamma$, reduces to Pythagoras when $\gamma = 90^\circ$.

## Calculus

$ \frac{\mathrm{d}}{\mathrm{d}x} x^n = n x^{n-1} \qquad \int x^n \,\mathrm{d}x = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) \qquad \int_a^b f(x)\,\mathrm{d}x = F(b) - F(a) $

$ e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!} \qquad \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e \qquad e^{i\pi} + 1 = 0 $

## Series and the golden ratio

$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \qquad \varphi = \frac{1 + \sqrt{5}}{2} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cdots}}} $

## Vectors, accents and alphabets

Vectors $\vec{v}$, unit vectors $\hat{\imath}$, means $\bar{x}$, time derivatives $\dot{x}$ and $\ddot{x}$, estimates $\hat{\theta}$, and the alphabets: $\mathbb{R}^n$, $\mathcal{L}$, $\mathfrak{g}$, $\mathbf{A}$, $\mathsf{T}$, $\mathtt{x}$.

$ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \qquad \mathbf{a} \cdot \mathbf{b} = \lVert \mathbf{a} \rVert \, \lVert \mathbf{b} \rVert \cos\theta \qquad \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} $

Greek, upper and lower: $\alpha, \beta, \gamma, \delta, \epsilon, \varepsilon, \theta, \lambda, \mu, \pi, \sigma, \varphi, \omega$ and $\Gamma, \Delta, \Theta, \Lambda, \Pi, \Sigma, \Phi, \Omega$.