126
D. Rocchesso: Sound Processing
angle measures 2 radians, and the 90
o
angle corresponds to /2 radians. The
main trigonometric functions are:
Sine sin = P Q
Cosine cos = OQ
Tangent tan = P Q/OQ
It is clear from fig. 7 and from the Pythagoras' theorem that, for any , the
identity
sin
2
+ cos
2
= 1
(21)
is valid.
The angle, considered positive if oriented anti clockwise, can be considered
the independent variable of trigonometric functions. Therefore, we can use Oc-
tave/Matlab to plot the main trigonometric functions, thus obtaining fig. 8.
These plots can be obtained as subplots of a same figure by the following Oc-
tave/Matlab script:
theta = [0:0.01:4*pi];
s = sin(theta);
c = cos(theta);
t = tan(theta);
subplot(2,2,1); plot(theta,s);
axis([0,4*pi,-1,1]);
grid; title('Sine of an angle');
xlabel('angle [rad]');
ylabel('sin');
% replot; % Octave only
subplot(2,2,2); plot(theta,c);
grid; title('Cosine of an angle');
xlabel('angle [rad]');
ylabel('cos');
% replot; % Octave only
subplot(2,2,3); plot(theta,t);
grid; title('Tangent of an angle');
xlabel('angle [rad]');
ylabel('tan');
axis([0,4*pi,-6,6]);
% replot; % Octave only
It is clear from the plots that the functions sine and cosine are periodic with
period 2, while the function tangent is periodic with period . Moreover, the
codomain of sine and cosine is limited to the interval [-1, 1], while the codomain
of the tangent takes values on all real axis. The tangent approaches infinity for
all the values of the argument that multiples of /2, i.e. in these points we have
vertical asymptotes.
As we can see from fig. 7, a complex number c, having magnitude and
argument , can be represented in its real and imaginary parts as
c = x + iy = cos + i sin .
(22)