You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.
The simplest signals are represented algebraically by functions of the form
where the constant omega determines the frequency. We can add constants A and delta as shown below. The constant A controls the amplitude of the signal -- we hear it as the loudness of the signal. The constant delta is called the phase and controls the phase of the signal -- basically when the signal was "turned on." We cannot "hear" the constant delta unless we combine two different signals with different phases. The module The Sound of Trigonometry discusses these signals in leisurely way.
It is often easier, mathematically, to look at functions of the form
The purpose of this module is to show that these two forms are equivalent -- that is, any function that can be expressed in the form
can also be expressed in the form
and vice versa. The key to changing back-and-forth between these two forms is the trigonometric identity.
Thus,
where
Write each of the following functions in the form
Check your answer by graphing both forms in your CAS window.
Conversely, we can start with the form
and write the same function in the form
as follows
where
Notice that since
and
we have
Write each of the following in the form
Check your work by graphing both forms in your CAS window.