# Closed form solution to sum of sine positive zero-crossings

• MHB
• Purplepixie
In summary, the problem is asking for a closed form solution to find the values of ##t## at which two conditions are simultaneously true for a given set of constants, but it is not solvable in general and requires specific values or relationships between the constants to find a solution.

#### Purplepixie

Hello,
I would like to know, if there's a closed form solution to the following problem:

Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what are the values of t at which two conditions are simultaneously true: (1) y = 0 and (2) the derivative dy/dt > 0. I am calling such points positive zero-crossing points, for want of a better name.

In other words, is there a closed form solution to the two simultaneous conditions:

sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) = 0
and
2.PI.a.cos(a.2.PI.t) + 2.PI.b.cos(b.2.PI.t) + 2.PI.c.cos(c.2.PI.t) > 0

Many thanks for any insights and assistance!

What do you mean by a closed form solution? Certainly one can figure out values. Pick ##a=b=c=1## for example. I think you will have to pick values to meet one condition and test the other with those values. One can easily do it graphically also.

I will write the problem in LaTeX, please correct me if I am wrong:

Given ##a, b, c##, define $$f(t)=\sin(2\pi at)+\sin(2\pi bt)+\sin(2\pi ct)$$Find ##t## such that ##f(t)=0## and ##\frac{df}{dt}(0)>0##

Hint: $\sin(x) + \sin(y) = 2 \sin(\frac{x+y}{2}) \cos(\frac{x-y}{2})$.

mathhabibi said:
I will write the problem in LaTeX, please correct me if I am wrong:

Given ##a, b, c##, define $$f(t)=\sin(2\pi at)+\sin(2\pi bt)+\sin(2\pi ct)$$Find ##t## such that ##f(t)=0## and ##\frac{df}{dt}(0)>0##
You mean
##\frac{df}{dt}(t)>0##?

• mathhabibi
There are an infinite number of solutions for ##t## given an arbitrary set of constants {##a,b,c##} however, I believe this problem is not solvable in closed form in general except for special cases when the constants have certain relationships.

• Greg Bernhardt