Question

Please help me answer this question

Answer 1

Answer:

• modulus: 3√2
• argument: -3π/4  (or 5π/4)

Step-by-step explanation:

The modulus is the magnitude of the complex number; the argument is its angle (usually in radians).

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rectangular form

The complex number can be cleared from the denominator by multiplying numerator and denominator by its conjugate:

  $\dfrac{-9+3i}{1-2i}=\dfrac{(-9+3i)(1+2i)}{(1-2i)(1+2i)}=\dfrac{-9+3i-18i-6}{1+4}=-3-3i$

polar form

The magnitude of this number is the root of the sum of the squares of the real and imaginary parts:

  modulus = √((-3)² +(-3)²) = 3√2

The argument is the arctangent of the ratio of the imaginary part to the real part, taking quadrant into consideration.

  arg = arctan(-3/-3) = -3π/4  or  5π/4 . . . . radians

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  modulus∠argument = (3√2)∠(-3π/4)