# Graphical Representation

The six trigonometric functions are represented graphically in the following images. The amplitude is on the y axis and the angle is on the x axis, in degrees. Often the angle is also represented in radians.

We have to keep in mind that these graphs are independent of time. The only independent variable is angle. They are actually a “photograph” in time of the known harmonic variable magnitudes.

For better visualization, two periods of each function are presented.

# Relation Between Trigonometric and Hyperbolic Functions

Trigonometric and Hyperbolic functions can be related between them using the following equations:

# Definition of Matrices

### Introduction

Matrices (or matrix in singular), is a compact notation method for element arrays. Often matrices are used with equations systems and vectors. The elements can then be easily treated.

Consider the following system of algebraic equations:

# Characteristic Equation, Eigenvalues

The characteristic equation of a square matrix A is:

where λ is a scaling factor.

This returns a polynomial, which is called the characteristic polynomial:

# Derived Forms of Matrices

### Matrix Transpose

In a matrix, if we change the position of the elements so that the rows become columns and the columns become rows, we get the transpose of the initial matrix.

# Determinants

### Defintion

Determinant is a number that is derived from a square matrix using an algorithm. If the determinant is other than zero, it means that the associated equations system has a unique solution or the associated vectors are linearly independent . The determinant of a square matrix is denoted as:

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