To convert an equation of the form
![\begin{displaymath}
f(\theta)=a\cos\theta+b\sin\theta
\end{displaymath}](h_385.gif) |
(1) |
to the form
![\begin{displaymath}
f(\theta)=c\cos(\theta+\delta),
\end{displaymath}](h_386.gif) |
(2) |
expand (2) using the trigonometric addition formulas to obtain
![\begin{displaymath}
f(\theta) = c\cos\theta\cos\delta-c\sin\theta\sin\delta.
\end{displaymath}](h_387.gif) |
(3) |
Now equate the Coefficients of (1) and (3)
so
![\begin{displaymath}
\tan\delta = - {b\over a}
\end{displaymath}](h_390.gif) |
(6) |
![\begin{displaymath}
a^2+b^2 = c^2,
\end{displaymath}](h_391.gif) |
(7) |
and we have
Given two general sinusoidal functions with frequency
:
their sum
can be expressed as a sinusoidal function with frequency
Now, define
![\begin{displaymath}
A\cos\delta \equiv A_1\cos\delta_1+A_2\cos\delta_2
\end{displaymath}](h_403.gif) |
(13) |
![\begin{displaymath}
A\sin\delta \equiv A_1\sin\delta_1+A_2\sin\delta_2.
\end{displaymath}](h_404.gif) |
(14) |
Then (12) becomes
![\begin{displaymath}
A\cos\delta \sin(\omega t)+A\sin\delta\cos(\omega t) = A\sin(\omega t+\delta).
\end{displaymath}](h_405.gif) |
(15) |
Square and add (13) and (14)
![\begin{displaymath}
A_2 = {A_1}^2+{A_2}^2+2A_1A_2\cos(\delta_2-\delta_1).
\end{displaymath}](h_406.gif) |
(16) |
Also, divide (14) by (13)
![\begin{displaymath}
\tan\delta = {A_1\sin\delta_1+A_2\sin\delta_2 \over A_1\cos\delta_1+A_2\cos\delta_2},
\end{displaymath}](h_407.gif) |
(17) |
so
![\begin{displaymath}
\psi = A\sin(\omega t+\delta),
\end{displaymath}](h_408.gif) |
(18) |
where
and
are defined by (16) and (17).
This procedure can be generalized to a sum of
harmonic waves, giving
![\begin{displaymath}
\psi = \sum_{i=1}^n A_i\cos (\omega t+\delta_i)= A\cos (\omega t+\delta),
\end{displaymath}](h_410.gif) |
(19) |
where
and
![\begin{displaymath}
\tan\delta = {\sum_{i=1}^n A_i\sin \delta_i\over \sum_{i=1}^n A_i\cos \delta_i}.
\end{displaymath}](h_414.gif) |
(22) |
© 1996-9 Eric W. Weisstein
1999-05-25