Some integer functions can readily be obtained by supplying integer arguments to continuous functions, so that for example Mod[x, 2] corresponds to Sin[ π x/2] 2 or (1 - Cos[ π x])/2, Mod[x, 3] ↔ 1 + 2/3(Cos[2/3 π (x - 2)] - Cos[2 π x/3]) Mod[x, 4] ↔ (3 - 2 Cos[ π x/2] - Cos[ π x] - 2 Sin[ π x/2])/2 Mod[x, n] ↔ Sum[j Product[(Sin[ π (x - i - j)/n]/ Sin[ π i/n]) 2 , {i, n - 1}], {j, n - 1}] (As another example, If[x > 0, 1, 0] corresponds to 1 - 1/Gamma[1 - x] .) … Page 147 showed how Sin[x] + Sin[ √ 2 x] has nested features, and these are reflected in the distribution of eigenvalues for ODEs containing such functions.

For Sin[a x] + Sin[b x] a more complicated sequence of substitution rules yields the analogous sequence in which -1/2 is inserted in each Floor .

Chords Two pure tones played together exhibit beats at the difference of their frequencies—a consequence of the fact that Sin[ ω 1 t] + Sin[ ω 2 t]  2 Sin[1/2( ω 1 + ω 2 ) t] Cos[( ω 1 - ω 2 ) t] With ω ≃ 500 Hz , one can explicitly hear the time variation of the beats if their frequency is below about 15 Hz, and the result is quite pleasant.

The formulas for local curvature as a function of arc length for each set of pictures are as follows: 1 (circle); s (Cornu spiral or clothoid); s 2 ; 1/Sqrt[s] (involute of circle); 1/s (logarithmic or equiangular spiral); 1/s 2 ; Exp[-s 2 ] ; Sin[s] ; s Sin[s] .

Sin[ ω t] corresponds to a pure tone. … FM synthesis functions such as Sin[ ω (t+ a Sin[b t])] can be made to sound somewhat like various musical instruments, and indeed were widely used in early synthesizers.

Artificial radio signals In current technology radio signals are essentially always based on carriers of the form Sin[ ω t] with frequencies ω /(2 π ) . When radio was first developed around 1900 information was normally encoded using amplitude modulation (AM) s[t] Sin[ ω t] . In the 1940s it also became popular to use frequency modulation (FM) Sin[(1 + s[t]) ω t] , and in the 1970s pulse code modulation (PCM) (pulse trains for IntegerDigits[s[t], 2] ).

, h n 2 and h Prime[n] ( h irrational) and probably n Sin[n] .

In the specific case a = 4 , however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms Sin[2 t ArcSin[ √ x ]] 2 (1 - Cos[2 t ArcCos[1 - 2 x]])/2 (1 - ChebyshevT[2 t , 1 - 2x])/2 where these follow from functional relations such as Sin[2x] 2  4 Sin[x] 2 (1 - Sin[x] 2 ) ChebyshevT[m n, x]  ChebyshevT[m, ChebyshevT[n, x]] For a = 2 it also turns out that there is a complete formula: (1 - (1 - 2 x) 2 t )/2 And the same is true for a = -2 : 1/2 - Cos[(1/3) ( π - (-2) t ( π - 3 ArcCos[1/2 - x]))] In all these examples t enters essentially only in a t .

This yields a chord such as Play[Evaluate[Apply[Plus, Flatten[Map[Sin[1000 # t] &, N[2 1/12 ]^Position[list, 1]]]]], {t, 0, 0.2}] A sequence of such chords can sometimes provide a useful representation of cellular automaton evolution.

In the pictures below, the n th point has position ( √ n {Sin[#], Cos[#]} &)[2 π n GoldenRatio] , and in such pictures regular spirals or parastichies emanating from the center are seen whenever points whose numbers differ by Fibonacci[m] are joined.