Sine and Cosine Phasors

Cuthbert Nyack

Sinusoids can be represented as rotating vectors.
A cosine can be represented as 2 rotating vectors as shown below:-

cos w t = ½ { e ^jwt + e ^-jwt }

e ^jwt is a counterclockwise rotating vector (orange) while e ^-jwt is a clockwise rotating (cyan) vector. The resultant is shown by the horizontal red line (½ size) and the vertical red shows the corresponding point on the cosine.

The sine is similar to the cosine with a phase shift as shown below:-

sin w t = j ³ ½ { e ^jwt - e ^-jwt }

The factor of j³ means that the clockwise rotating vector starts at -270º (90º) and the counterclockwise rotating vector starts at +270º(-90º).

When activated the following gif image show how the applets should appear.

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