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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