### Trigonometry: Trigonometric functions

### Signals in the time domain

From now on we use the independent variable \(t\) because we usually will interpret it as time. A function \(s(t)\) of \(t\) is then called a **(time) signal.**

A signal of the form \[s(t)=A \sin(2\pi f t+\varphi)\] is called an **alternating signal** (also known as **sinusoid)** with **amplitude** \(A\), **frequency** \(f\) and **phase** \(\varphi\).

We will always assume that \(A\) and \(f\) are positive numbers and call this formula the *standard form.*

The amplitude determines the size of the oscillations.

The frequency determines the number of oscillations in the time interval [0,1].

The unit of frequency, that is, the number of periods per second, is hertz **(Hz).**

The phase is related to a horizontal translation

(to the left for positive \(\varphi\), to the right for negative \(\varphi\)).

The number \(\displaystyle T=\frac{1}{f}\) is the **period** of the alternating signal, because \(T\) is the smallest positive integer with the property that \[s(t+T)=s(t)\] for all \(t\).

In the figure to the right we have draen the graph of the sinusod defined by \(s(t)=3\sin\left(\tfrac{1}{2}\pi+\tfrac{1}{2}\pi\right)\). We denote herein the amplitude (\(3\)), the period (\(4\)) en and the horizontal translation (\(-1\)) of a sine graph that has, in this exampe, a phase of \(\tfrac{1}{2}\pi\).

Alternating signal as a sum of sine and cosine By making use of the addition formulas any alternating signal can also be written in the form \[s(t)=a \sin(2\pi f t)+b \cos(2\pi f t)\] Here\[a=A\cos(\varphi)\quad\text{and}\quad b=A\sin(\varphi)\]

Conversely, any signal of the form \[s(t)=a \sin(2\pi f t)+b \cos(2\pi f t)\] can be written in standard form. You must determine \(A\) and \(\varphi\) so that at least \[A=\sqrt{a^2+b^2}\quad\text{and}\quad \tan(\varphi)=\frac{\sin(\varphi)}{\cos(\varphi)}=\frac{b}{a}\] The appropriate phase \(\varphi\) is fixed modulo \(2\pi\).