Thursday, 14 August 2008

Addition Of Trigonometric Functions

Trigonometric functions are like pure numbers. There are even and odd numbers.
Likewise trigonometry has even and odd functions.

For more information on Even and Odd function of trigonometry, click here.

What happen when cosine is added to sine?

Since cosine function is even, and sine is odd, their summation will result in an overall odd function. This is similar to number concept.

Therefore, we can conclude that:
  • cosine + sine ==> Odd function

  • cosine + cosine ==> Even function

  • sine + sine ==> Odd function


But what if a fundamental sine is added to its harmonics? Example sin A + sin 2A.

This will still produce an odd function, though the shape of the resultant wave will not be "sine" anymore.

The next tempting question to ask is :

"What happens when more of the harmonics are added up?".

Let us do an example with sin A and looking at the waveforms.

Here, I shall utilise only odd numbered harmonics to demonstrate, as the cumulative effect will be more apparent and easier to understand.

The first wave is done up using sin A + sin 3A.

The second wave is done using sin A + sin 3A + sin 5A + sin 7A + sin 9A + sin 11A.




What you see in the diagram are 3 waveforms. Note the last waveform (green colored) is the fundamental sine shown for reference.

The first (blue colored) wave is "sin A + sin 3A". Observe the shape of this wave. It is steeper than the reference sinA wave.

The second (purple colored) wave is 6 odd harmonics added up. It is a lot more harmonics than the first. Look carefully at its shape. It has the steepest gradient at its edge.

What does this "more harmonics" has on the overall shape?

Answer: The more harmonics the trigonometric function has, the shaper or steeper the wave can be done up. Every function has to have lots more of harmonics to has a faster rate of change.

The more harmonics a function has, the less the response time of the function.

Therefore, when you see a wave that has many sharp edges or steep slopes, you can easily deduce it to have many strong harmonics.

This is the principle of trigonometry adddition.

This understanding has many engineering applications:

- Response time for motor control (fast recovery)

- Data transmission decision (rate of change of signal bits)

- Fidelity of audio signal (bandwidth of system, decision for good repreoduction)

Trigonometry addition is a simple operation, but it has great impact in many fields.

Can you now see the significance of understanding maths and their usefulness?

:) Maths is WonderFUL :)
:) Maths is InteResTing :)
.

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1-+cos(angle)=2sin(半角)平方 2cos(半角)平方 1-+sin(angle)=(sin角半-+cos半角)^two
347.learnbank.com.tw/highschool/highschool_1_ m2.php
沈赫哲數學
賴樹聲物理數學工程數學