@Carl Love I replace 3, 5, 7 by 3, 4, 5, I got "Warning, expecting only range variable t in expressions [abs(cx)*cos(t), abs(cx)*sin(t)] to be plotted but found name cx
Warning, data could not be converted to float Matrix".

@Carl Love Thank you very much.

@vv I use Maple 2018. I got

@Kitonum I think, we can remove 

eq := sin(9*x-(1/3)*Pi) = sin(7*x-(1/3)*Pi):

@Carl Love I think
restart;
a := 9*x-(1/3)*Pi:
b := 5*x-(1/6)*Pi:
solve(a = 2*Pi*k+b, x);
solve(a = 2*Pi*k+Pi-b, x);

@Mariusz Iwaniuk 
How to use your code with 
sin(9*x-(1/3)*Pi) = sin(5*x-(1/6)*Pi)?

@nm Yes, k is an Integer. I agree with you.

@tomleslie Thank you vey much.

@tomleslie Thank you very much. A small question. Why I get error when I use nops(evalf(sol)) ?

@Carl Love How can I get the exact of the numbers a, b, c?
By hand, I can prove that, the line Delta passing the point DD and perpendicular to the plane (ABC). And then,

maxS = Distance(A, DD)+Distance(B, DD)+Distance(C, DD);

My code get 

[9.37769297305868754, [a = .801783725149603, b = .534522484467394, c = .267261242390448]]

@Joe Riel Thank you very much.

@mmcdara Because I choose polynomial degree 10, then when tends to plus or minus infinity, f(x) always tend to infinity. It didn't fit with figure. 
I think, we need to choose a function that is suitable for the drawing. 

@tomleslie I like your answer. It closes with my picture. I copied your code to LaTeX and used piecewise. It looks good (except that the dots are maybe not exactly at the positions of the local extrema). Can you repair positions of the local extrema?

@mmcdara With the given figure, When x  tends towards  +infinity, then f(x) tends towards  +infinity, too?

@tomleslie Thank you very much.