Question 1.17: Graph each of the following: (a) 6(cos 240° + isin 240°), (b...
Graph each of the following: (a) 6(cos 240° + isin 240°), (b) 4e^{3\pi}=5, (c) 2e^{-\pi i}/4
(a) 6\left(\cos 240^{\circ}+i \sin 240^{\circ}\right)=6 \operatorname{cis} 240^{\circ}=6 \operatorname{cis} 4 \pi / 3=6 e^{4 \pi i / 3} can be represented graphically by OP in Fig. 1-26.
If we start with vector OA, whose magnitude is 6 and whose direction is that of the positive x axis, we can obtain OP by rotating OA counterclockwise through an angle of 240°. In general, r e^{i \theta} is equivalent to a vector obtained by rotating a vector of magnitude r and direction that of the positive x axis, counterclockwise through an angle \theta.
(b) 4 e^{3 \pi i / 5}=4(\cos 3 \pi / 5+i \sin 3 \pi / 5)=4\left(\cos 108^{\circ}+i \sin 108^{\circ}\right) is represented by OP in Fig. 1-27.
(c)2 e^{-\pi i / 4}=2\{\cos (-\pi / 4)+i \sin (-\pi / 4)\}=2\left\{\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right\}
This complex number can be represented by vector OP in Fig. 1-28. This vector can be obtained by starting with vector OA, whose magnitude is 2 and whose direction is that of the positive x axis, and rotating it counterclockwise through an angle of -45° (which is the same as rotating it clockwise through an angle of 45°).
