General Sinusoid Properties.

Cuthbert Nyack
The applet here illustrates some of the basic properties of Sinusoids. f(t) = a cos(wt + f) + b sin(Wt + q).

Fn = 1 shows how a sinusoid with phase f1 can be represented as a sum of a sine(red) and a cosine(cyan). f1 must be changed to see the different components.
Fn = 2 to 5 show special cases of Fn = 1.

Fn = 6 shows how adding a cosine with amplitude 'a' and a sine with amplitude 'b' combine to result in a sinusoid. The magnitude and phase of the resulting sinusoid is shown at the bottom in blue.
Fn = 7 to 10 show special cases of Fn = 6.

Fn = 11 shows how 1 period of a cosine can be represented as the sum of 2 counterrotating vectors in the complex plane ie cos(wt) = ½ (ejwt + e-jwt).
'a' must be changed to see the vectors at different times.
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.
Fn = 12 to 15 shows special cases of Fn = 11.

Fn = 16 shows how 1 period of a sine can be represented as the sum of 2 counterrotating vectors in the complex plane ie sin(wt) = 1/(2j) (ejwt - e-jwt).
'a' must be changed to see the vectors at different times. The imaginary resultant is shown horizontally here.
Fn = 17 to 20 shows special cases of Fn = 16.

Fn = 21 shows how a sines and cosines are affected by changing their phase.
The top plots show standard sine and cosines. The bottom plots show their phase shifted counterparts. the phase shift is set by f1 and may be +ve or -ve corresponding to advanced or delayed sinusoids.
Fn = 21 to 25 shows special cases of Fn = 21.

Fn = 26 shows how a sines and cosines are affected by Time Shifting.
The top plots show standard sine and cosines. The bottom plots show their Time shifted counterparts. the Time shift is set by T and may be +ve or -ve corresponding to advanced or delayed sinusoids.
Fn = 27 to 30 shows special cases of Fn = 26.

Fn = 31 illustrates the orthogonality of 2 sines.
The applet shows sin(nwt)sin(mwt) (w refers to angular frequency omega) for different n and m. The +ve part of product function is shown shaded red, and the -ve part shaded blue. The red and blue areas cancel out. When n = m the product is always positive and averages 1/2. the product averages zero whenever n is not equal to m.
The width of the applet corresponds to the fundamental period T. Product must be averaged over T(= 2 p/w) (or nT) to get zero. The Average value is shown on the applet. For m not equal to n, the error (< 2E-16) shown is numerical error. Fn = 32 to 35 shows special cases of Fn = 31.

Fn = 36 illustrates the orthogonality of a sine and a cosine.
Fn = 37 to 40 shows special cases of Fn = 36.

Fn = 41 illustrates the orthogonality of 2 cosines.

Fn = 42 to 45 shows special cases of Fn = 41.

Fn = 46 illustrates the time domain addition of 2 sinusoids.
Fn = 47 to 51 shows what is considered to be 'consonant' frequencies.
Fn = 52 , 53 show imperfect 'consonant' frequencies.
Fn = 54 to 57 shows what is considered to be 'dissonant' frequencies.

Fn = 58 illustrates the time domain addition of 3 sinusoids.
Fn = 59 , 60 show 3 frequency 'chords'.

Fn = 61 illustrates the Lissajous figures. sin(n1wt) is applied to the H axis and sin(n2wt) + sin(n3wt) applied to the V axis.
Fn = 62 to 77 shows special cases of Fn = 61.

Fn = 78 shows a parametric plot of the functions x = a sin(t), y = b cos(t).
Fn = 79 shows the case where a = b. The result is a circle and this is the reason why sines and cosines are referred to as circular functions.
Fn = 80 and 81 shows other cases of Fn = 78.

Fn = 82 shows a parametric plot of the functions x = a sinh(t), y = a cosh(t).
The result is a hyperbola and is the reason why sinh and cosh are referred to as hyperbolic functions.
Fn = 83 to 85 shows other cases of Fn = 82.

One of the consequences of the circular property of sinusoids is that a wide variety circle based patterns can easily be deduced from elementary arithmetic operations with sinusoids.
Fn = 86 to 538 shows a tiny subset of the possibilities.

The following gif images show some of the circle based patterns.