Saturday, 16 August 2008

How To Sketch Trigonometric Graph

There are times when we are required to sketch trigonometric waveform given the mathematical expression. By understanding the mathematical principle explained below, you can be rest assured that sketching the graphs is a breeze. Here we go .....

Trigonometric function relies on 3 parameters to define its shape (or graph).

They are :

  1. Amplitude
  2. Period
  3. Phase shift

Thus, the trigonometric expression can be represented by y = a sin (bA - c), where y and A are the variables.

In this post, a step-by-step explanation is done.

Let's take a reference sine wave as starting point.

The reference graph of y = sin A is shown below.

Diag 1: Sine wave (reference)

Here the amplitude is "1", period is 3600 and phase shift is 00.
The period is defined as the total angle taken to complete one full cycle.
Phase shift is the amount of angle shift from the starting 00.

Now, what happen when we increase the amplitude by 2.
Y = 2 sin A

Diag 2: Amplitude increased

Here, we can see that the height or amplitude of the reference sine wave is increased from 1 to 2, with all else remaining the same.

Next, let us change the period parameter and see what happen ==> y = sin 2A.

NOTE: The formula to compute the numerical value of period of a sine or cosine wave is
PERIOD = 3600 / b, where "b" is the coefficient next to the angle A in "sin bA".

Diag 3: The compressed sine wave y = sin 2A.

What happened in Diag 3 is that when we increased the sin A to sin 2A, the period to have a complete cycle reduces by half or by 1/ b.

If the trigonometric expression is y = sin 3A, that is b = 3, the new period would then be reduced to 1/3 the original ( or 360 / 3 = 120 degree).

Let's change the last parameter, the Phase Shift.
y = sin (A - 200) ==> an addition of "- 200".

What is the sine wave like?

Diag 4: The shifted reference sine wave by positive 200.

NOTE: The formula to calculate the phase shift is

PHASE SHIFT = - c / b,

with "c" and "b" being the parameters represented in the generic sine wave expression
y = a sin (bA - c).

Another point and a common mistake by learners, is to take the number "c" as the actual phase shift angle. It is wrong!

The number "c" has to be divided by the angle coefficient "b" before the actual phase shift can be obtained. And note the sign "-" also!

Look closely at the angle of shift. The sine wave is shifted right (or positvely) by 200, even though the expression is "- 200".

Why so?

The y = sin (A - 200) means that to achieve the same amplitude as the reference sine wave at 00, A has to be at +200. Therefore A has to be at + 200 away from the origin. Understand?

Let's look at the final composite sine wave with all parameters changed.
y = 2 sin (2A - 200).

Diag 5: Composite sine wave y = 2 sin (2A - 200).

From diag 5, we can see that the phase shift is 100 only, with the period reduced by half. The amplitude is also increased to 2 as in the trigonometric equation.

I have finished my explanation of getting the sine wave sketched given a trigonometric expression. You can see that it is not that difficult. What is needed is the 3 simple steps of defining the amplitude , period and phase shift, taking care of the sign.

As long as we understand the meaning of the parameters in defining trigonometry graphs, the principle of sketching them is straight-forwardly easy. Do you agreed?



kimo said...

thank you i understand alot now, but what if you have somthing like y=2sin(3pi t/2)

EeHai said...

In the post I have utilised "degree" to be my domain of operation (x-axis). However, if you like, you may change it to "radian" unit. The relation between degree and radian is 360 degree = 2(pi).
For the question regarding 3pi t/2, the domain is a bit confusing. Is it "t" or in radian?
Normally, another form similar is 2(pi)ft, where f = frequency of rotation.

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