11. Linear combinations

For some purposes it is important to know that any linear combination of sine waves of the same period or frequency but different phase shifts is also a sine wave with the same period or frequency, but a different phase shift. This is useful in sinusoid data fitting, because the measured or observed data are linearly related to the a and b unknowns of the in-phase and quadrature components basis below, resulting in a simpler Jacobian, compared to that of c and φ.

Sine and cosine

In the case of a non-zero linear combination of a sine and cosine wave (which is just a sine wave with a phase shift of π/2), we have

${\displaystyle a\sin x+b\cos x=c\cdot \sin(x+\varphi )\,}$

where

${\displaystyle c={\sqrt {a^{2}+b^{2}}},\,}$

and (using the atan2 function)

${\displaystyle \varphi =\operatorname {atan2} \left(b,a\right).}$

Arbitrary phase shift

More generally, for an arbitrary phase shift, we have

${\displaystyle a\sin x+b\sin(x+\theta )=c\sin(x+\varphi )\,}$

where

${\displaystyle c={\sqrt {a^{2}+b^{2}+2ab\cos \theta }},\,}$

and

${\displaystyle \varphi =\operatorname {atan} {\frac {b\,\sin \theta }{a+b\cos \theta }}.}$

More than two sinusoids

The general case reads^{[citation needed]}

${\displaystyle \sum _{i}a_{i}\sin(x+\theta _{i})=a\sin(x+\theta ),}$

where

${\displaystyle a^{2}=\sum _{i,j}a_{i}a_{j}\cos(\theta _{i}-\theta _{j})}$

and

${\displaystyle \tan \theta ={\frac {\sum _{i}a_{i}\sin \theta _{i}}{\sum _{i}a_{i}\cos \theta _{i}}}.}$

See also Phasor addition.