Our goal here is to understand the identity

where and

Before deriving this identity, use the following interactive graph to plot and

What are the characteristics of the sum function?


The graphs suggest that we search for a formula for the sum that is just a single sinusoidal term. One way to express the general form of a sinusoidal function with a given frequency is

where R is the amplitude and is the phase angle.

We use the trigonometric identity

with and . Then

How should R and be chosen so that for all t?

 

The amplitude R should be positive so that

However, you need to be careful in solving for . To determine an explicit expression for , consider all four possible cases for the signs of A and B.

Case 1: A and B are both positive.
First, draw the picture.

Based on this figure, what is ?

Case 2: A is negative and B is positive
First draw the picture.

Remember that arctan(x) has a range of

Based on this figure what is ?

Do Case 3 (A is positive and B is negative) and Case 4 (A and B are both negative) on your own.

For another way to express the sum of two sinusoidal functions, go to practice.