Adding a Cosine and a Sine.
Cuthbert Nyack
A sine has a phase of 0 at t = 0 and a cosine has a phase of
p/2 at t = 0.
Adding a sine and a cosine can result in a sinusoidal
signal with any phase. This can be represented as :-
a sin (w t + f) =
(a cos f ) sin w t +
(a sin f ) cos w t.
a cos (w t - (p/2 - f)) =
(a cos (p/2 - f) ) cos w t +
(a sin (p/2 - f) ) sin w t.
=
(a cos f ) sin w t +
(a sin f ) cos w t.
The applet below shows a cosine (cyan) and a sine (red) added to produce a
sinusoid in (magenta). As the phase of the resulting magenta sinusoid
is changed the vector diagram in the middle of the
plot shows how the sine and cosine are added vectorially.
The amplitude of the resultant is kept constant.
This is similar to resolving a vector in the xy plane in terms of its x and
y components, in this case sine and cosine.
When activated the following gif image show how the applet should appear.
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COPYRIGHT © 2005 Cuthbert Nyack.