What Is Euler's Formula (e^iθ)?

Euler's formula states

$$e^{i\theta}=\cos\theta+i\sin\theta.$$

It represents a complex number with modulus $1$ and argument $\theta$, connecting exponential form, trigonometric form, and points on an Argand diagram.

How Euler's Formula Works

On the unit circle, the point at angle $\theta$ from the positive real axis has Cartesian coordinates $(\cos\theta,\sin\theta)$. Interpreting the horizontal coordinate as the real part and the vertical coordinate as the imaginary part gives

$$z=\cos\theta+i\sin\theta=e^{i\theta}.$$

Multiplying by a positive modulus $r$ changes the distance from the origin without changing the angle. Therefore, the general Euler form of a non-zero complex number is

$$z=re^{i\theta}=r(\cos\theta+i\sin\theta).$$

To convert from Cartesian form, calculate

$$r=|z|=\sqrt{a^2+b^2}$$

and determine $\theta=\arg(z)$, checking the quadrant from the signs of $a$ and $b$.

For example, when $\theta=\frac{2\pi}{3}$,

$$e^{\frac{2\pi i}{3}}=\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)=-\frac{1}{2}+\frac{\sqrt{3}}{2}i.$$

Setting $\theta=\pi$ gives

$$e^{i\pi}+1=0.$$

This is Euler's identity, which is a special case of Euler's formula rather than another name for it.

Euler form also makes multiplication and division efficient. If $z_1=r_1e^{i\alpha}$ and $z_2=$, then

$$z_1z_2=r_1r_2e^{i(\alpha+\beta)}.$$

This means moduli multiply while arguments add; for division, moduli divide and arguments subtract. For example,

$$(2e^{i\pi/6})(3e^{i\pi/4})=6e^{i5\pi/12}.$$

This compact rule is the basis for applying De Moivre's theorem to powers and roots in AHL 1.14. Geometrically, multiplying rotates a complex number through the new argument and scales its distance from the origin.

A common misconception is writing $e^{i\theta}=\cos\theta+\sin\theta$. This is incorrect because the sine component lies on the imaginary axis, so the factor $i$ is essential.

IB Exam Technique

This is IB Mathematics AA HL content from AHL 1.13: Polar and Euler form. Paper 1 questions usually require exact trigonometric values and analytic conversion. On Paper 2, a GDC may help calculate the modulus and argument, but the correct form and quadrant must still be shown. Examiners expect radians unless degrees are specified and may link Euler form to multiplication, division, powers, or roots of complex numbers.