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This chapter was intentionally meant to be as short as possible, so that you (the reader) can start with the real content (starting in Chapter 2) swiftly. To navigate in this book, you can use the menu button in the upper left corner, and click on links for chapters, sections, figures, etc.

In the next section, we will introduce our book's basic notation, which will

This section describes the notation used in this book. The reader may skip this now, and check back when needed, but it is a quick and simple read, so it may be preferable to do it before diving into the other chapters.

Note that everything is numbered in this book. This is to make it simpler to discuss the different equations, expressions, etc with another person. That is, you can always refer to Equation 3.12, which means that the equation/expression is in Chapter 3, and it is the 12:th one there.

A number, in the common sense, may be positive, negative, it may have decimals, and it may even be rational or irrational. The set of all such numbers are denoted by $\R$, and the numbers are said to be real numbers. We often use the term a

The absolute value of a scalar, $k$, is denoted $\abs{k}$, and is defined as

\begin{equation} \abs{k} = \left\{ \begin{matrix} k, & \mathrm{if\ } k \geq 0\\ -k, & \mathrm{if\ } k < 0 \end{matrix} \right. , \end{equation} | (1.1) |

A set is a small collection of, for example, integer numbers. A set with the numbers $1$, $2$, and $4$ is denoted: $\{1,2,4\}$. When it is desirable to have a variable, $i$, for example, that can take on any of the members in a set, then we write: $i\in\{1,2,4\}$.

Real numbers can often take on values in a certain range. As an example if we know $x$ can take on any value from zero to one, including zero and one, we can write $x \in [0, 1]$. A soft parenthesis marks that the end point is not included. As examples,

\begin{gather} x \in [-1, 2) \ \ \ \mathrm{denotes} \ \ \ -1 \leq x < 2 \\ \theta \in (-\frac{\pi}{2}, \frac{\pi}{2}] \ \ \ \mathrm{denotes} \ \ \ -\frac{\pi}{2} < \theta \leq \frac{\pi}{2}. \end{gather} | (1.2) |

This section provides a summary of some useful trigonometry concepts that will be used in this book, and that said, it is by no means a thorough treatment of the subject.

The cosine, sine, and tangent functions are always good to have fresh in mind. Refer to Figure 1.1. For right triangles, the following holds:

\begin{align} \cos \theta &= \frac{a}{c}, \\ \sin \theta &= \frac{b}{c}, \\ \tan \theta &= \frac{\sin \theta}{\cos \theta} = \frac{b/c}{a/c} = \frac{b}{a}. \end{align} | (1.3) |

Next, a set of useful trigonometric identities will be presented. The trigonometric unity is

\begin{equation} \cos^2\theta + \sin^2\theta = 1. \end{equation} | (1.4) |

\begin{equation} c^2 = a^2 + b^2 - 2ab\cos \gamma. \end{equation} | (1.5) |

\begin{equation} \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c}. \end{equation} | (1.6) |

The inverse functions for the cosine and sine are also important to know about. For example, $\theta = \arcsin a$ means that $\sin \theta = a$. The function $\arcsin a$ can take on values in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Likewise $\theta = \arccos a$ means that $\cos \theta = a$, and $\arccos a \in [0, \pi]$.

Note however that just because $\sin \theta = a$ we cannot conclude that $\theta = \arcsin a$. As an example, $\sin 4\pi = 0$, but $\arcsin 0 = 0$, not $4\pi$. In fact $\sin \theta = a$ has an infinite number of solutions, and $\arcsin$ is only the inverse to $\sin$ within the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$. However we can list all the solutions to $\sin \theta = a$ as $\theta = \arcsin(a) \pm 2 k \pi$, where $k = 0, 1, 2, 3, \ldots$ and so on.

At this point, it is time to start with the first real chapter of this book. Click here to go to Chapter 2, or navigate using the menu at the left top corner.