Wednesday, 7 January 2009

Trigonometric Form of Complex Number

Trigonometric form of the complex number combines algebra with trigonometry.

Z = r(cosA + jsinA)

In Trigonometry:(examples)

cos(450)= 0.707, sin(450) = 0.707, and
cos(-450) = 0.707, sin(-450) = -0.707,

cos(1350) = -0.707, sin (1350 = 0.707, and
cos(-1350) = -0.707, sin (-1350) = -0.707

The above characteristics showed that "cosine" has less impact to the overall angular interpretation of the complex number, Z.

What do I mean?

If Z = 3(cos450 + j sin 450), we know the angle to be +450.

However, if Z = 3(cos450 - j sin 450), what is the angle?
"+450" as in the cosine term, or "-450" as for the sine term.

Looking at the trigonometric examples using positive and negative angles, the sign of the computed values having the same angle for the cosine operation remained.

Rather, the imaginary term of sinA has more information regarding the actual value of the angle.
This is so since "sine" can cause sign change to the computed value with positive or negative angles.

Thus to quickly identify the angle from the trigonometric form, we have just to look at the imaginary or sine portion of the complex number.
The cosine part will not reveal the correct answer.

An understanding of trigonometrical principles, especially the quadrant concepts, has to be strong in order to handle complex number studies.

:D Do not be overly fearful of maths. It is interesting, if you follow the ideas behind it.

:) ....