Chapter 9.3, Problem 26AYU

To determine

The rectangular form of the complex number $2e^{i\cdot \frac{5\pi }{6}}$.

Answer to Problem 26AYU

Solution:

The rectangular form of the complex number $2e^{i\cdot \frac{5\pi }{6}}$ is $-\sqrt{3}+i$.

Explanation of Solution

Given information:

The complex number $2e^{i\cdot \frac{5\pi }{6}}$.

Explanation:

Consider the complex number $z=2e^{i\cdot \frac{5\pi }{6}}$.

Polar co-ordinate of this complex number is $\left(r,\theta \right)=\left(2,\frac{5\pi }{6}\right)$.

By writing the given complex number in polar form,

$z=2e^{i\cdot \frac{5\pi }{6}}=2\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$

Substitute $\cos \frac{5\pi }{6}=-\frac{\sqrt{3}}{2}$ and $\sin \frac{5\pi }{6}=\frac{1}{2}$ in the complex number $2\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$,

$2\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)=2\left(-\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)$

Simplifying,

$\Rightarrow -\sqrt{3}+i$

Therefore, the rectangular form of the complex number $2\left(\cos \frac{5\pi }{6}+i\sin \frac{5\pi }{6}\right)$ is $-\sqrt{3}+i$.