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:
In this function A is the amplitude and f is the frequency. The angle of the function can be expressed in degrees or in radians. The relationship between them is 360 degrees equal to 2π radians.
In the next images we can see how amplitude and frequency alter the graph for a value 2 for the amplitude and 3 for the frequency.
Image 1: y=1sin(x)
Image 2: y=2sin(x)
Image 3: y=1sin(3x)
Image 4: y=2sin(3x)
We can understand from image 1 and 2 that the graph repeat itself every 360 degrees (or 2π radians) when the frequency has value one Hz. This is called period and its symbol is T. When the frequency is higher, for example has a value of three, then the graph in 360 degrees repeat itself three times. In this case the period is: 360/3=120 degrees.
In electrical and electronic engineering, is very common to insert the time variable in the trigonometric functions. This allows us to know the value of a magnitude, like voltage or current, at a certain value of time. In this case the equation is:
Where A is the amplitude, f is the frequency and t is the time. Usually when we have the time in the equation, we express the angle in radians, because it is easier and “cleaner” in writing. In the next graph we can see an example voltage of 5V amplitude and 10Hz frequency.
Image 5: V=5sin(2π10t)
We have to keep in mind that in this graph the x axis is the time and not the angle. This expresses the variable magnitude (i.e. the voltage) over time. Time is the only independent variable. It is like we are standing in the sea and having closed our eyes, we are using our hands to feel the ups and downs (amplitude) of the sea surface over time.
The product 2πf is usually written as ω. The equation is then:
The relationship between the period T and the frequency f of the trigonometric functions is T=1/f.
Sometimes an angle θ is added or subtracted to the product ωt. This is a phase shift. The equations are then like:
When the angle θ is added to ωt the graph is shifted left by θ radians and when it is subtracted the graph is shifted right by θ radians.