Find the modulus and argument of: z= ((1 +j2)^2 * (4−j3)^3) / ((3 +j4)^4 * (2−j)^3)

Question

Ans: (√5)/25,−2.035

Please show step by step

Answer 1

One way of solving this problem is to simplify the expression and then transform into polar coordinates.

It is better if we transform each of the expressions into polar coordinates and then calculate the modulus and the argument separately.

 

Let z₁ =1+2j,

z₂ = 4-3j,

z₃ = 3+4j,

z₄ = 2-j.

 

We will denote the modulus of the z₁ to z₄ with R₁ to R₄ and will denote the arguments with the corresponding indexes ∝_{1 to} ∝₄.

 

We will calculate R₁ and ∝₁ first.

R₁=√(1²+2²)

R₁=√5.

∝₁= tan^-1(2/1).

 

We will keep these in this form, so we can finally use a calculator, like TI 30 or similar to calculate the value of the angle.

 

The same way we calculate the remaining values:

R₂=√(4²+(-3)²)

R₂=5.

∝₂=-tan^-1(3/4)

This is because the complex number is in quadrant 4, with positive real part and negative imaginary part.

 

R₃=√(3²+4²)

R₃=5.

∝₃=tan^-1(3/4).

 

R₄=√(2²+(-1)²)

R₄=√5.

∝₄=-tan^-1(1/2)

 

We are ready to calculate R and ∝.

Let write z first again:

z = [z₁² • z₂³] / [z₃4 • z₄³ ].

 

The modulus R will follow the formula for z:

R =[R₁² • R₂³ ]/[R₃⁴ • R₄³ ]

R =[√5² • 5³ ]/[5⁴ • √5 ³].

 

You can simplify this answer to verify the value.

 

The value of the argument ∝ follows different path. It is the multiplied sum of the corresponding tan^-1 by the power of the complex number z_i.

 

∝ = 2• ∝₁ + 3•∝₂ - 4•∝₃ - 3•∝₄

∝ = 2• tan^-1(2/1) + 3•(-tan^-1(3/4)) - 4•tan^-1(4/3) - 3•(-tan^-1(1/2)).

 

Use a calculator to verify the answers.

 

This problem is long and tedious. I suggest you break similar problems into peaces and make clear notes of what you are doing. Organizing this type of problems and using a consistent notation will help you trace and check your answers.