Question

Express the following numbers in the form x + iy:

${\displaystyle {\text{e}}^{\frac{5\pi }{6}\text{i}}}$

Answer 1

z = re^iθ = ${\displaystyle {\text{e}}^{\frac{5\pi }{6}\text{i}}}$

∴ r = 1, θ = ${\displaystyle \frac{5\pi }{6}}$

Polar form of z = r(cos θ + i sin θ)

= ${\displaystyle 1(\cos \frac{5\pi }{6}+\text{i}\sin \frac{5\pi }{6})}$

= ${\displaystyle \cos (\pi -\frac{\pi }{6})+\text{i}\sin (\pi -\frac{\pi }{6})}$

= ${\displaystyle -\cos \frac{\pi }{6}+\text{i}\sin \frac{\pi }{6}}$

= ${\displaystyle -\frac{\sqrt{3}}{2}+\text{i}\cdot \frac{1}{2}}$

= ${\displaystyle -\frac{\sqrt{3}}{2}+\frac{1}{2}\text{i}}$