What Happen When Sine and Cosine Are Added

Why do we care about the addition or subtraction of trigonometric functions?

(The basic Trigonometric functions are sine and cosine).

Answer:

Most signal, including our voice or speech, can be represented by the addition and subtraction of trigonometric functions.

Example of a signal: sin 2A + sin 4A - cos 2A - cos 4A

Another example: cos A + sin A - cos 3A - sin3A

The signals above are different in their profile.

But most signal can be mathematically represented, by choosing the appropriate functions and their harmonics, .

This is the principle of speech processing.

To modify the profile of any signal, what we have to do is to add or subtract the relevant components of the functions.

Components?

This is the term used to identify the element of frequency and amplitude in a trigonometric function.

We know that a trigonometric function can be written as K cos 2 p f t, with the

"f" being the frequency of the function and

"K" being the amplitude (or amount) of the function.

To view more information about trigonometry and their relation to angle and frequency, visit this post here.

More details:

sin p + cos p = 0 + (-1) = -1 (at angle p )

sin p + 2 cos p = 0 + 2(-1) = -2 (at angle p )

2 sin p/2 - cos p/2 = 2(1) - 0 = 2 (at angle p/2 )

From the 3 examples of above, you can see that we can manage the outcome of the resultant amplitudes by boosting the relevant components within the expressions.
The frequency factors are also manageable by controlling the angle (indirectly the frequency, since angle ==> 2 p f t).

Therefore by monitoring or adjusting the amplitude and frequency elements of the components, we can analyse or manipulate the shape of most signals. This is an application of trigonometric maths.

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