Question

if z= 4+4√3 find modulus and amplitude of z​

Answer 1

Step-by-step explanation:

Let z = x + iy where x and y are real numbers and i = √(-1). Then the non-negative square root of (x2 + y2) is known as the modulus or absolute value of z. Modulus or absolute value of z is denoted by |z| and read as mod z.

Hence if z = x + iy, then |z| = |x+iy| = +√x2 + y2.

For example, if z = -3 + 4i then, |z| = |-3 + 4i |= √(-3)2 + 42 = 5.

AMPLITUDE (OR ARGUMENT) OF A COMPLEX NUMBER:

Let z = x + iy where x and y are real numbers and i = √(-1) and x2 + y2 ≠ 0, then the value of θ for which the equations x = |z| cosθ ........(1) and y = |z| sin θ .......(2) are concurrently satisfied is named as the amplitude or argument of z and is denoted by Amp z or Arg z.

Equations (1) and (2) are satisfied for infinitely many values of θ, any of these infinite values of θ is the value of amp z. However, the unique value of θ lying in the interval -π< θ ≤ π and satisfying equations (1) and (2) is known as the principal value of arg z and it is denoted by arg z or amp z. Or in other words argument of a complex number means its principal value.

Since, cos(2nπ + θ)= cos θ and sin(2nπ + θ)= sin θ (where n is an integer), hence

Amp z = 2nπ + amp z where -π < amp z ≤ π.

Answer 2

Answer:

Hlo

Here is your answer

The modulus of a complex number is found by adding the squares of the real part and the imaginary part's coefficients and taking their square root.

Here, it will be

The amplitude of the complex number is the angle it forms with the x-axis on a graph

In this question, the slope of line on which the number lies is -4/-4 = 1

tan x = 1

x = 45° or 225°

Since both coefficients are negative, the point lies in the third quadrant.

So x = 225°

Hope it helps

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