Question

The principal amplitude of ${\left(\sin \left(40°\right)+i\cos \left(40°\right)\right)}^5$

Answer 1

Find the principal amplitude:

From De Moivre's Theorem we know that for $n$ be a positive integer

$\cos \left(n\theta \right)+i\sin \left(n\theta \right)={\left(\cos \left(\theta \right)+i\sin \left(\theta \right)\right)}^n$

$\therefore {\left(\sin \left(40°\right)+i\cos \left(40°\right)\right)}^5=i^5{\left(\cos \left(40°\right)+\frac{1}{i}\sin \left(40°\right)\right)}^5$

$$\begin{gathered} ={\left(-1\right)}^{2}i{\left(\cos \left(40°\right)+\frac{i}{i·i}\sin \left(40°\right)\right)}^5\left[\because i^2=-1\right] \\ =i{\left(\cos \left(40°\right)-i\sin \left(40°\right)\right)}^5\left[\because i^2=-1\right] \\ =i\left(\cos \left(5\times 40°\right)-i\sin \left(5\times 40°\right)\right)\left[\because \cos \left(n\theta \right)+i\sin \left(n\theta \right)={\left(\cos \left(\theta \right)+i\sin \left(\theta \right)\right)}^n\right] \\ =i\left(\cos \left(200°\right)-i\sin \left(200°\right)\right) \\ =i\cos \left(200°\right)-{\left(i\right)}^2\sin \left(200°\right) \\ =\left(\sin \left(200°\right)+i\cos \left(200°\right)\right)\left[\because i^2=-1\right] \\ =\sin \left(180°+20°\right)+i\cos \left(180°+20°\right) \\ =-\sin \left(20°\right)-i\cos \left(20°\right)\left[\because \sin \left(\pi +\theta \right)=-\sin \left(\theta \right),\cos \left(\pi +\theta \right)=-\cos \left(\theta \right)\right] \\ =\cos \left(90°+20°\right)-i\sin \left(90°+20°\right)\left[\because \sin \left(\frac{\pi }{2}+\theta \right)=\cos \left(\theta \right),\cos \left(\frac{\pi }{2}+\theta \right)=-\sin \left(\theta \right)\right] \\ =\cos \left(110°\right)-i\sin \left(110°\right) \\ =\cos \left(-110°\right)+i\sin \left(-110°\right)\left[\because \sin \left(-\theta \right)=-\sin \left(\theta \right),\cos \left(-\theta \right)=\cos \left(\theta \right)\right] \end{gathered}$$

For a complex number $z=\cos \left(\theta \right)+i\sin \left(\theta \right)$; $\theta$ is called principal amplitude

Hence, the principal amplitude is $-110°$