91Ó°ÊÓ

Determine, in polar form (a) \(\left[2 \angle 35^{\circ}\right]^{5}\) (b) \((-2+j 3)^{6}\) (a) \(\begin{aligned}\left[2 \angle 35^{\circ}\right]^{5}=2^{5} \angle\left(5 \times 35^{\circ}\right), \\ \text { from de Moivre's theorem } \\\=& \mathbf{3 2} \angle \mathbf{1 7 5}^{\circ} \\\\(-2+j 3)=\sqrt{\left[(-2)^{2}+(3)^{2}\right]} \angle \tan ^{-1} \frac{3}{-2} \end{aligned}\) (b) \(\begin{aligned}(-2+j 3)=&\left.\sqrt{\left[(-2)^{2}+(3)^{2}\right.}\right] \angle \tan ^{-1} \frac{3}{-2} \\\=& \sqrt{13} \angle 123.69^{\circ}, \text { since }-2+j 3 \\ \text { lies in the second quadrant } \end{aligned}\) $$ \begin{aligned} (-2+j 3)^{6} &=\left[\sqrt{13} \angle 123.69^{\circ}\right]^{6} \\ &=(\sqrt{13})^{6} \angle\left(6 \times 123.69^{\circ}\right), \end{aligned} $$ by de Moivre's theorem $$ \begin{aligned} =& 2197 \angle 742.14^{\circ} \\ =2197 \angle 382.14^{\circ}(\text { since } 742.14\\\ \left.\quad \equiv 742.14^{\circ}-360^{\circ}=382.14^{\circ}\right) \\ =2197 \angle 22.14^{\circ}\left(\text { since } 382.14^{\circ}\right.\\\ \equiv &\left.382.14^{\circ}-360^{\circ}=22.14^{\circ}\right) \\ \text { or } 2197 \angle{22}^{\circ} \mathbf{8}^{\prime} \end{aligned} $$

Convert \(7.2 \mathrm{e}^{j 1.5}\) into rectangular form. \(7.2 \mathrm{e}^{j 1.5}=7.2 \angle 1.5 \mathrm{rad}\left(=7.2 \angle 85.94^{\circ}\right)\) in polar form \(=7.2 \cos 1.5+j 7.2 \sin 1.5\) \(=(0.509+j 7.182)\) in rectangular form

Determine the locus defined by \(|z-2|=3\), given that \(z=x+j y\) If \(\begin{aligned} z=x+j y, \quad \text { then } \quad|z-2|=|x+j y-2| \\\ &=|(x-2)+j y|=3 \end{aligned}\) On the Argand diagram shown in Figure \(23.3\), \(|z|=\sqrt{x^{2}+y^{2}}\) Hence, in this case, \(|z-2|=\sqrt{(x-2)^{2}+y^{2}}=3\) from which, \((x-2)^{2}+y^{2}=3^{2}\) From Chapter \(14,(x-a)^{2}+(y-b)^{2}=r^{2}\) is a circle, with centre \((a, b)\) and radius \(r\). Hence, \((x-2)^{2}+y^{2}=3^{2}\) is a circle, with centre \((2,0)\) and radius 3 The locus of \(|z-2|=3\) is shown in Figure 23.6.