**Complex Numbers and
Angle Addition Formulas**

**By: Kelli Parker**

** **

As
a math student, I have struggled with remembering the angle addition formulas
for sine and cosine. The two
formulas are so similar, it can be very easy to mix them up, but it would be
deadly to do so on an exam. A
recent assignment for a math education class helped me realize that I can use
multiplication of complex numbers to help remember these two formulas!

First,
recall the sine and cosine angle addition formulas:

Certainly,
there are beautiful trigonometric proofs we could do, using identities and
substitution. Instead of using
trig, however, we can follow the pattern of integrated mathematics and use a
completely different topic, complex numbers, to explain and remember these
formulas.

Now
for a little review of complex number ÒanatomyÓ:

Take
a complex number, *z* = *a + bi*. We
can think of *z*
like a vector, with Òx and yÓ components, which are in this case, Òreal and
imaginaryÓ. We call *a* the ÒrealÓ component because
it has no *i*
term in it. Looking at the plot of *z* above in the Real-Complex plane, the *a* term is the component on the
horizontal, or real, axis. Our *b* is like a scalar which we multiply
to *i*. We call *bi* the imaginary component of
our complex number because of the *i* term; we see the imaginary component on the
vertical, or ÒiÓ axis in our plot above.

We
can also represent *z* in polar form. We
do this by first finding the angle between the real axis and *z*, which we call . Allow *z* to be a radius of a unit
circle. The coordinates for the
endpoint of *z*
that lies on the unit circle can be obtained as on the normal trig unit circle:
by bringing in sine and cosine.
Recall that this is done by creating a right triangle with hypotenuse *z*, as in the plot below. So we can now use our trig
relationships and see that the sides of this triangle make up the components of
*z*. Where before we had *a* for the real part, we now
can use cos; similarly for the imaginary part, instead of using *b*, we can use sin. Since it is a
rare occasion that *z* would have a length (or modulus) of 1, we will call its length
*r*. Since our cosine and sine relationships
were based on a hypotenuse of 1, we can just multiply by *r* to get the correct value.

So now our complex number is represented in polar
form:

Now
that weÕve looked at the polar form of *z*, letÕs choose two more complex numbers, *x* and *y*. We will represent them in the first way: *x* = *c+di *and *y = m+ni*. LetÕs look at the product of *x* and *y*:

Notice
that we combine the real components in the first 2 terms (those terms not
containing *i*)
and the imaginary components in the last 2. We are able to substitute -1 for *i*^{2} because this
is a property of *i*.

Next,
letÕs consider *x *and
*y *in their
polar form. For ease of notation,
we will call the angle for *x* angle(x), and we will call the angle for *y* angle(y), noting and
understanding that these are not equivalent with the numbers themselves. We will also take an *r* value of 1, for simpler
calculation. So we have *x = *cos(x)+ *i*sin(x), and

*y
*= cos(y)+ *i*sin(y). LetÕs look at the product in polar
form:

We
have something that looks like a combination of the two angle addition
formulas, but is still the product of 2 complex numbers. It corresponds exactly to the general
form product we took above; the corresponding values of the two forms would be *c
= *cos(x), *d* = sin(x), *m* = cos(y), and *n* = sin(y). So if we substitute the cosine and sine
values into the general form product we found above, we obtain the following:

So
when we combine the real components, we get cos(x)cos(y) – sin(x)sin(y). Remember that in polar form, the cosine
value corresponds to the real component.
So for the cosine part of the complex number, we have obtained the
cosine angle addition formula! The
same is true if we look at the imaginary part, the coefficient of *i*:

cos(x)sin(y)+
sin(x)cos(y), which is the sine angle addition formula, since sine corresponds
to the coefficient of the imaginary part of complex numbers!

Next
time memory fails and you canÕt remember which formula is which, think about
multiplying 2 complex numbers, and then substitute the sine and cosine values
of the polar form into the product, and after you group reals and imaginaries
together, youÕll have your angle addition formulas.