To find the Amplitude or Argument of a complex number let us assume that, a complex number z = x + iy where x > 0 and y > 0 are real, i = √1 and x\(^{2}\) + y\(^{2}\) ≠ 0; for which the equations x = z cos θ and y = z sin θ are simultaneously satisfied then, the value of θ is called the Argument (Agr) of z or Amplitude (Amp) of z.
From the above equations x = z cos θ and y = z sin θ satisfies infinite values of θ and for any infinite values of θ is the value of Arg z. Thus, for any unique value of θ that lies in the interval  π < θ ≤ π and satisfies the above equations x = z cos θ and y = z sin θ is known as the principal value of Arg z or Amp z and it is denoted as arg z or amp z.
We know that, cos (2nπ + θ) = cos θ and sin (2nπ + θ) = sin θ (where n = 0, ±1, ±2, ±3, .............), then we get,
Amp z = 2nπ + amp z where  π < amp z ≤ π
Algorithm for finding
Argument of z = x + iy
Step I: Find the value of tan\(^{1}\) \(\frac{y}{x}\) lying between 0 and \(\frac{π}{2}\). Let it be α.
Step II: Determine in which quadrant the point M(x, y) belongs.
If M (x, y) belongs to the first quadrant, then arg (z) = α.
If M (x, y) belongs to the second quadrant, then arg (z) = π  α.
If M (x, y) belongs to the third quadrant, then arg (z) =  (π  α) or π + α
If M (x, y) belongs to the fourth quadrant, then arg (z) = α or 2π  α
Solved Examples to find the Argument or Amplitude of a complex number:
1. Find the argument of the complex number \(\frac{i}{1  i}\).
Solution:
The given complex number \(\frac{i}{1  i}\)
Now multiply the numerator and denominator by the conjugate of the denominator i.e., (1 + i), we get
\(\frac{i(1 + i)}{(1  i)(1 + i)}\)
= \(\frac{i + i^{2})}{(1  i^{2}}\)
= \(\frac{i  1}{2}\)
=  \(\frac{1}{2}\) + i ∙ \(\frac{1}{2}\)
We see that in the zplane the point z =  \(\frac{1}{2}\) + i ∙ \(\frac{1}{2}\) = (\(\frac{1}{2}\), \(\frac{1}{2}\)) lies in the second quadrant. Hence, if amp z = θ then,
tan θ = \(\frac{\frac{1}{2} }{ \frac{1}{2}}\) = 1, where \(\frac{π}{2}\) < θ ≤ π
Thus, tan θ = 1 = tan (π \(\frac{π}{4}\)) = tan \(\frac{3π}{4}\)
Therefore, required argument of \(\frac{i}{1  i}\) is \(\frac{3π}{4}\).
2. Find the argument of the complex number 2 + 2√3i.
Solution:
The given complex number 2 + 2√3i
We see that in the zplane the point z = 2 + 2√3i = (2, 2√3) lies in the first quadrant. Hence, if amp z = θ then,
tan θ = \(\frac{2√3 }{2}\) = √3, where θ lying between 0 and \(\frac{π}{2}\).
Thus, tan θ = √3 = tan \(\frac{π}{3}\)
Therefore, required argument of 2 + 2√3i is \(\frac{π}{3}\).
`11 and 12 Grade Math
From Amplitude or Argument of a Complex Number to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.