period of the sum of sines?

Assuming that m, n > 0.

(If m and/or n are negative, use sin(-t) = - sin t to remove the negative signs.)

The period of sin(nx) is 2π/n, and the period of sin(mx) is 2π/m.

==> The period of sin(nx) + sin(mx) is lcm(2π/n, 2π/m), if this exists.

Examples:

The period of sin(2x) + sin(3x) is lcm(2π/2, 2π/3) = 2π.

There is no period for sin(x) + sin(πx), because lcm(2π, 2) does not exist.

I hope this helps!

sin nx + sin mx = 2 sin {(nx+mx)/2} cos{(nx-mx)/2}