Editing Phasor (section)

==Definition==
A real-valued sinusoid with constant amplitude, frequency, and phase has the form:

:<math>A\cos(\omega t + \theta),</math>

where only parameter <math>t</math> is time-variant.  The inclusion of an [[imaginary part|imaginary component]]:

:<math>i \cdot A\sin(\omega t + \theta)</math>

gives it, in accordance with [[Euler's formula]], the factoring property described in the lead paragraph:

:<math>A\cos(\omega t + \theta) + i\cdot A\sin(\omega t + \theta) = A  e^{i(\omega t + \theta)} = A e^{i \theta} \cdot e^{i\omega t},</math>

whose real part is the original sinusoid.  The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids.  Furthermore, all the mathematics can be done with just the phasors <math>A e^{i \theta},</math> and the common factor <math>e^{i\omega t}</math> is reinserted prior to the real part of the result.

The function <math>Ae^{i(\omega t + \theta)}</math> is an ''[[analytic representation]]'' of <math>A\cos(\omega t + \theta).</math> Figure&nbsp;2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a ''phasor'',<ref>{{cite book |last1=Singh |first1=Ravish R |title=Electrical Networks |date=2009 |publisher=Mcgraw Hill Higher Education |isbn=978-0070260962 |page=4.13 |chapter=Section 4.5: Phasor Representation of Alternating Quantities}}</ref> as we do in the next section.