# Definitions of Trigonometric Functions

We have the right angle triangle ABC. From this orthogonal triangle the trigonometric functions are defined as follows:

We have the right angle triangle ABC. From this orthogonal triangle the trigonometric functions are defined as follows:

Trigonometric functions can be expressed in exponential form using Euler’s number e=2,71.

The trigonometric functions are defined from the orthogonal triangular as a ratio of its sides. In order to interpret them geometrically we will use the unit circle. The unit circle consists of a circle of radius length of one unit, which is set on the centre of the Cartesian coordinates. We will also consider a line of angle θ in respect to axis x, that begins from the origins of the Cartesian system and intersects the unit circle at the point A.

In image 1 we can see how sine, cosine and tangent are geometrically interpreted. Sine is the length of the y axis from its origin up to the projection of the point A on the y axis, cosine is the length of the x axis from its origin up to the projection of the point A on the x axis and tangent is the length of the tangent of the unit circle at the point A, from that point, up to the point that it intersects the axis x.

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.

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

Sum and Difference of Trigonometric Equations

The six trigonometric functions have two main characteristics, the amplitude and the frequency. The amplitude describes the height and the depth of the function, while the frequency describes how fast it is. We will use the sine function to illustrate the examples and the rest of the functions are similar. A sine function could be: