MTH403 Assignment 1 Solution Fall 2026

This assignment covers fundamental concepts in complex numbers and their properties as presented in the first eight lectures of the course. Students are required to solve five mathematical problems involving polar forms, loci in the complex plane, triangle inequalities, complex exponential functions, and trigonometric identities. You must show all necessary steps for each calculation and use proper mathematical formatting for your final submission.

1. 1

   Question 1

   5 marks

   Express the complex number z = 1 - i√3 in polar form. You must include the calculation of both the modulus and the argument in your steps.

2. 2

   Question 2

   5 marks

   Find the locus of points in the complex plane that satisfy the condition -1 ≤ Re(z) < 1, where z is defined as x + iy.

3. 3

   Question 3

   5 marks

   Determine the upper and lower bounds of |z1 + z2| using the triangle inequality. The given values are z1 = -√3 + i and z2 = -i.

4. 4

   Question 4

   5 marks

   Given z = x + iy where x and y are real numbers, show that the absolute value of the exponential function |e^(z^2+3z)| equals e^(x^2-y^2+3x).

5. 5

   Question 5

   5 marks

   Prove the provided trigonometric identity using De Moivre's theorem. The expression involves powers of complex trigonometric terms resulting in cis(-20θ).

What to submit

One Word file using Math Type or Equation Editor for all mathematical symbols.

Topics covered

Complex numbers Polar form Complex plane Triangle inequality Complex exponential functions De Moivre's theorem Trigonometric identities