Question

Express this complex number in trigonometric form (5/2)*sqrt(3) + (5/2)*i

Answer 1

Question #27826

Express this complex number in trigonometric form $(5/2)*\operatorname{sqrt}(3) + (5/2)*i$.

Solution. A complex number $z$ is in trigonometric form if $z = r(\cos\varphi + \sin\varphi)$, where $r = |z|$.

Denote $z = \frac{5}{2}\sqrt{3} + \frac{5}{2}i$. Then $z = 5\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right)$. It follows immediately that $\cos\varphi = \frac{\sqrt{3}}{2}$, $\sin\varphi = \frac{1}{2}$ and $r = 5$ and so $= \frac{\pi}{6}$. Thus, $z = 5\left(\cos\frac{\pi}{6} + \sin\frac{\pi}{6}\right)$.

Answer. $5\left(\cos \frac{\pi}{6} + \sin \frac{\pi}{6}\right)$.