## Thursday, 14 August 2008

### Multiplication of Trigonometric Functions

What is it like to multiply (cos A) with (cos B)? It can happen!

Can the answer be expressed in another form that is easier to understand?

Before we can answer that, let us slightly deviate to another related topic of angle summation (difference).

We know of these Trigonometric Identities:

1. sin (A + B) = sin A cos B + sin B cos A

2. sin (A - B) = sin A cos B - sin B cos A

3. cos (A + B) = cos A cos B - sin A sin B

4. cos (A - B) = cos A cos B + sin A sin B

Making use of the identities above, we can express multiplication of 2 trigonometric functions into a simpler form consisting of angle addition (or subtraction).

Let's take back the same question at the beginning of this post.

cos A cos B = ?

From the above identities (3) and (4), we can see that they consisted of the product term "cos A cos B".

Therefore, we will make use of them to simplify cos A cos B into an understandable form.

To get rid of the other product terms within the identities (3) and (4), that is, term sinA sinB, what we need to do is to add up the 2 identities (3) and (4).

cos (A + B) + cos (A - B) <==> 2 cos A cos B

What does the result means?

It means that the product of 2 functions is the SUM of another similar functions with angle addition and subtraction.

Confused? Hope not.

A diagram of the Frequency Spectrum will reveal the simplicity.

Diag: Frequency Spectrum of cos A cos B

From the diagram, you can see that the multiplication operation on 2 cosine terms resulted in only 2 simple vertical frequency components. And that's it, nothing complex.

Likewise, sin A sin B also produces 2 frequency components.

sin A sin B = (1/2) [ cos (A - B) - cos (A + B)]

And,

sin A cos B = (1/2) [sin (A + B) + sin (A - B)]

cos A sin B = (1/2) [sin (A + B) - sin (A - B)]

All these multiplications produce only 2 frequency components which are added (or subtracted) and is simpler mathematically.

(Note: You can see the effectiveness of using Frequency method presentation to simplify explanation here.)
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