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Given also that Z3 is a complex number such that |Z3|=5 and the argument of Z3 is rac2pi3. Express Z3 in polar form.

Mathematics

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It seems there was some confusion with the numbering in our previous exchanges, as the image you sent contains questions 1 through 5. I have already solved the following questions from this image:

  • Question 1 (i) a) and b) (previously referred to as "Question 8 i) a)" and "Question 8 i) b)")
  • Question 1 (ii) (previously referred to as "Question 8 ii)")
  • Question 2
  • Question 3
  • Question 4 (i) (identical to Question 2)
  • Question 4 (ii) (previously referred to as "Question 7")

Let's continue by solving Question 5 from the image.


Question 5: Given also that Z3Z_3 is a complex number such that ∣Z3∣=5|Z_3|=5 and the argument of Z3Z_3 is 2π3\frac{2\pi}{3}. Express Z3Z_3 in polar form.

  1. Identify the modulus and argument of Z3Z_3.

    The modulus is given as ∣Z3∣=5|Z_3| = 5.
    The argument is given as arg⁡(Z3)=2π3\arg(Z_3) = \frac{2\pi}{3}.

  2. Write Z3Z_3 in polar form.

    The polar form of a complex number zz with modulus rr and argument θ\theta is z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta).
    Substitute the given values for Z3Z_3:

    Z3=5(cos⁡(2π3)+isin⁡(2π3))Z_3 = 5\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)

    The complex number Z3Z_3 in polar form is 5(cos⁡(2π3)+isin⁡(2π3))\boxed{5\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)}.

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