Chapter 6 Elementary Functions

Section 6.5 Trigonometric Functions

6.5.2 Sine Function

In Module 5 the trigonometric functions were introduced in Section 5.6 on right triangles, for example by the relation

sin(α)= opposite side hypotenuse ,

and explained using the unit circle. Starting from this definition of sin(α) we arrive at the sine function if we declare the angle α to be the independent variable of a function named sin. This can be illustrated by means of a family of right triangles ABC that are inscribed the unit circle, i.e. a circle with radius r=1, in a certain way.

If we start with the angle α= 0 , corresponding to a triangle degenerated to a line segment, then the length of the line segment BC equals 0. If we now rotate the point B counterclockwise around the circle, then the angle α - and sin(α) as well - increases until for α= 90 a maximum value ( sin( 90 )=1) is reached. Afterwards, the angle α continues to increase while sin(α) is starting to decrease again. For α= 180 , the triangle ABC is again degenerated to a line segment and sin( 180 )=0. If α further increases, the triangle "flips downwards" and the line segment BC is oriented parallel to the negative vertical line ( y-axis), hence, its length is negative. For α= 270 the maximum negative value occurs, before α approaches 0 again. At α= 360 the game starts again.
The figure above shows the graph of the sine function

sin:  { [-1;+1] αsin(α) .

In contrast to the discussion so far, the angle α is plotted on the horizontal axis ( α-axis) in radian measure, used more commonly in this context, and not in degree measure.
Let us specify some of the most relevant properties of the sine function:
  • The sine function is defined on the entire set of real numbers . Hence, Dsin =. In contrast, the range only consists of the interval between -1 and +1, including these two endpoints: Wsin =[-1;+1].
  • After certain measures, the graph of the since function repeats its shape exactly; in this context this is called the periodicity of the sine function. The period of the sine function is 360 or 2π. Mathematically, this relation can be expressed as


Just a quick look at the graph of the simple sine function suggests that we could use this function for the description of wave phenomena. However, to be able to use the full potential of the sine function a few parameters will be introduced. For example, the amplitude of the sine function can be amplified or damped by a so-called amplitude factor A, the frequency of the oscillation can be affected by a frequency-like factor a, and the entire path of the graph can be shifted to the left or to the right by a shifting constant b. Thus, the general sine function has the following form:

f:  { [-A;+A] xf(x)=Asin(ax+b) .

Example 6.5.1
Let us consider a simple pendulum. A small, heavy weight swings freely in the gravitational field of the Earth at a long, very thin cord that is, for example, fixed at the ceiling of a (high) room. Under certain idealised conditions and for small values of the displacement angle φ from the rest position (the vertical axis), the relation between the angle φ and the independent variable t, i.e. the time, is described by a general sine function:

φ(t)=Asin( 2π T t+b).

Here, T denotes the so-called period of the pendulum, i.e. the period of time, required by the pendulum for a full oscillation.