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and

z −1 = 1 exp( j − )θ (6.41)
r

from which we can deduce Euler’s equations:

exp( ) exp(θ +j − ) θ j
cos( )θ = (6.42)
2

and

exp( ) exp(θ −j − ) θ j
sin( )θ = (6.43)
j 2

Example 6.3
Use MATLAB to generate the graph of the unit circle in the complex plane.

Solution: Because all points on the unit circle are equidistant from the origin
and their distance to the origin (their modulus) is equal to 1, we can generate
the circle by plotting the N-roots of unity, taking a very large value for N. This
can be implemented by executing the following script M-file.

N=720;
z=exp(j*2*pi*[1:N]./N);
plot(z)
axis square

In-Class Exercises
Pb. 6.20 Using the exponential form of the n-roots of unity, and the expres-
sion for the sum of a geometric series (given in the appendix), show that the
sum of these roots is zero.
Pb. 6.21 Compute the following sums:
a. 1 + cos(x) + cos(2x) + … + cos(nx)
b. sin(x) + sin(2x) + … + sin(nx)
c. cos(α) + cos(α + β) + … + cos(α + nβ)
d. sin(α) + sin(α + β) + … + sin(α + nβ)
Pb. 6.22 Verify numerically that for z = x + jy:

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