Exercise 429Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

Modulus and argument

#modulus#argument#trigonometric form

Statement

Find the modulus and argument of $z=1+i\sqrt3$ , then write $z$ in trigonometric form.

Hint

$|z|=\sqrt{a^2+b^2}$ , $\arg(z)$ determined by $\cos\theta=a/|z|$ , $\sin\theta=b/|z|$ .

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Modulus. $|z|=\sqrt{1+3}=2$ .

Argument. $\cos\theta=1/2$ , $\sin\theta=\sqrt3/2$ , so $\theta=\pi/3$ .

Trigonometric form. $z=2(\cos\frac\pi3+i\sin\frac\pi3)=2e^{i\pi/3}$ .

Conclusion. $|z|=2$ , $\arg(z)=\pi/3$ (mod $2\pi$ ), $z=2e^{i\pi/3}$ .