What is Euler's formula?
e^{iθ} = cos θ + i sin θ. It links the exponential function to trigonometry, so a complex number of modulus r and argument θ is r·e^{iθ} = r(cos θ + i sin θ).
// maths › Advanced Applications
Euler's Formula & de Moivre
Apply Euler's formula e^{iθ} = cos θ + i sin θ and de Moivre's theorem to write and power complex numbers, plotted on an Argand diagram, in degrees or radians.
e^{iθ} = cos θ + i sin θ; (r(cosθ+isinθ))ⁿ = rⁿ(cos nθ + i sin nθ)
A mind behind this: Leonhard Euler 1707–1783
Angle unit
Frequently asked questions
What is de Moivre's theorem?
(r(cos θ + i sin θ))ⁿ = rⁿ(cos nθ + i sin nθ): raise the modulus to the power and multiply the argument by n.
Can you show a worked example?
For z = 2(cos 40° + i sin 40°), z³ = 2³(cos 120° + i sin 120°) = 8(cos 120° + i sin 120°). In radians the argument is 40° ≈ 0.6981 rad and the result argument is 120° ≈ 2.0944 rad.
What is the Argand diagram?
A plane where the horizontal axis is the real part and the vertical axis the imaginary part, so a complex number is a point — and z and zⁿ can be drawn as rays from the origin.
Where is this used?
Deriving multiple-angle identities, finding the n roots of a complex number, AC circuit phasors, and the Fourier transform.