Question B.1: Express the following complex numbers in polar and exponenti...

Notice that we have deliberately chosen these complex numbers to fall in the four quadrants, as shown in Fig. B.2.

(a) For z_{1}=6+j 8 (1st quadrant),

r_{1}=\sqrt{6^{2}+8^{2}}=10, \quad \theta_{1}=\tan ^{-1} \frac{8}{6}=53.13^{\circ}

Hence, the polar form is 10∠53.13° and the exponential form is 10 e^{j 53.13^{\circ}} .
(b) For z_{2}=6-j 8 (4th quadrant),

r_{2}=\sqrt{6^{2}+(-8)^{2}}=10, \quad \theta_{2}=360^{\circ}-\tan ^{-1} \frac{8}{6}=306.87^{\circ}

so that the polar form is 10∠306.87° and the exponential form is 10 e^{j 306.87^{\circ}} . The angle \theta_{2} may also be taken as −53.13°, as shown in Fig. B.2, so that the polar form becomes 10∠− 53.13° and the exponential form becomes 10 e^{-j 53.13^{\circ}} .
(c) For z_{3}=-6+j 8 (2nd quadrant),

r_{3}=\sqrt{(-6)^{2}+8^{2}}=10, \quad \theta_{3}=180^{\circ}-\tan ^{-1} \frac{8}{6}=126.87^{\circ}

Hence, the polar form is 10∠126.87° and the exponential form is 10 e^{j 126.87^{\circ}} .
(d) For z_{4}=-6-j 8 (3rd quadrant),

r_{4}=\sqrt{(-6)^{2}+(-8)^{2}}=10, \quad \theta_{4}=180^{\circ}+\tan ^{-1} \frac{8}{6}=233.13^{\circ}

so that the polar form is 10∠233.13° and the exponential form is 10 e^{j 233.13^{\circ}} .