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.