When the graphs intersect f(x)=g(x) so 600sin(2(pi)/3(x-0.25))+1000=600sin(2(pi)/7(x))+500.
600(sin(2(pi)/3(x-0.25))-sin(2(pi)/7(x)))+500=0.
The 7 fundamental values of x satisfying this equation are to be found at the end of this answer.
Trig identities:
sin(p)=sin(p+2n(pi)) and cos(p)=cos(p+2m(pi)), where n and m are integers,
so sin(2(pi)(x-0.25)/3)=sin(2(pi)(x-0.25)/3+2n(pi))=sin(2(pi)(x+3n-0.25)/3) and
cos(2(pi)x/7)=cos(2(pi)x/7+2m(pi))=cos(2(pi)(x+7m)/7). For g(x) the value of the function repeats for x, x+7, x+14, etc.; and for f(x) it's x, x+3, x+6 etc. The repetition of intersections of f(x) and g(x) occur for x, x+21, x+42, etc., where x is a solution of the combined equation determining the intersection points.
sin(A-B)=sinAcosB-cosAsinB and sin(A+B)=sinAcosB+cosAsinB.
sin(A+B)-sin(A-B)=2cosAsinB.
If X=A+B and Y=A-B, X+Y=2A so A=(1/2)(X+Y) and X-Y=2B so B=(1/2)(X-Y)
sinX-sinY=2cos((X+Y)/2)sin((X-Y)/2). Using these identities X=2(pi)/3(x-0.25), Y=2(pi)x/7.
(X+Y)/2=(pi)((x-0.25)/3+x/7)=(pi)(7x-1.75+3x)/21=(pi)(10x-1.75)/21
(X-Y)/2=(pi)((x-0.25)/3-x/7)=(pi)(7x-1.75-3x)/21=(pi)(4x-1.75)/21
600(sinX-sinY)+500=0 so sinX-sinY=-5/6=2cos((pi)(10x-1.75)/21)sin((pi)(4x-1.75)/21)
cos((pi)(10x-1.75)/21)sin((pi)(4x-1.75)/21)=-5/12.
Solutions for x: 1.67126, 3.03768, 7.82582, 9.27714, 14.13103, 15.28939, 16.90941, 22.67126,...
There are 7 fundamental values for x and a series can be built on each one by adding (or subtracting) 21 (LCM of 3 and 7). The intersections of the sine waves repeat the earlier pattern indefinitely.
Addendum
The table below shows some points on the graphs of f(x) and g(x) for the purposes of comparison and to show roughly when f(x) intersects g(x). The comparison column f(x)~g(x) shows whether f(x) is less than or greater than g(x). Where there is a change from < to >, or vice versa, an intersection point exists between the values of x listed. The table shows two whole cycles of f(x) (0<x<6) and one cycle of g(x) (0<x<7) with selected values of x to show the trend of each function. The table can be extended by including selected values of 7<x<21 to show 7 complete cycles of f(x) and 3 complete cycles of g(x), after which the two functions return to their initial phasing.
Comparison of f(x) and g(x)
| x |
f(x) |
g(x) |
f(x)~g(x) |
| 0 |
700 |
500 |
> |
| 0.25 |
1000 |
633.5 |
> |
| 0.5 |
1300 |
760.3 |
> |
| 0.75 |
1519.6 |
874.1 |
> |
| 1 |
1600 |
969.1 |
> |
| 1.25 |
1519.6 |
1040.6 |
> |
| 1.5 |
1300 |
1085 |
> |
| 1.75 |
1000 |
1100 |
< |
| 2 |
700 |
1085 |
< |
| 2.25 |
480.6 |
1040.6 |
< |
| 2.5 |
400 |
969.1 |
< |
| 2.75 |
480.6 |
874.1 |
< |
| 3 |
700 |
760.3 |
< |
| 3.5 |
1300 |
500 |
> |
| 4 |
1600 |
239.7 |
> |
| 4.5 |
1300 |
30.9 |
> |
| 5 |
700 |
-85 |
> |
| 5.5 |
400 |
-85 |
> |
| 6 |
700 |
30.9 |
> |
| 6.5 |
1300 |
239.7 |
> |
| 7 |
1600 |
500 |
> |