Amplitude or Argument of a Complex Number

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 z-plane 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 z-plane 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 

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