r = ones(1,4);
theta = [0 pi/2 pi 3*pi/2];
[x,y] = pol2cart(theta,r);
cla
plot(cos(range),sin(range),'k')
% Plot the unit circle
hold on
plot(x, y, 'ro')
% Plot the values of 1, i, -1, and -i
text(x(1)+0.05,y(1),'1')
% Add text labels
text(x(2),y(2)+0.1,'i')
text(x(3)-0.1,y(3),'-1')
text(x(4)-0.02,y(4)-0.1,'-i')
Knowing this, we can write the following expression for i:
i = cos 2k + 1/2 π + i sin 2k + 1/2 π .
Taking the nth root of both sides gives
i
1/n = cos 2k + 1/2 π + i sin 2k + 1/2 π
1/n
and by de Moivre's theorem we get
i
1/n = cos 2k + 1/2 π + i sin 2k + 1/2 π
1/n = cos
2k + 1/2 π
n
+ i sin
2k + 1/2 π
n
.
19 Live Scripts and Functions
19-82