Trigonometric and exponential form

Problem

Write $z = -1 + i$ in trigonometric and exponential form.

Solution

Compute the modulus and the argument. The modulus: $\rho = |z| = \sqrt{(-1)^2 + 1^2} = \sqrt 2.$ The point $(-1;1)$ lies in the second quadrant, so the argument is $\theta = \dfrac{3\pi}{4}$. Therefore: $z = \sqrt 2\left(\cos\tfrac{3\pi}{4} + i\sin\tfrac{3\pi}{4}\right) = \boxed{\sqrt 2\,e^{\,i\,3\pi/4}}.$

Links

Topics: Complex numbers
Concepts: Argument · Exponential form · Trigonometric form · Modulus
Methods: Complex numbers in polar form
Skills: Calculating · Using formulae
Exercise type: Evaluating expressions