# 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

## 1. What is a closed form solution?

A closed form solution is a mathematical expression that can be written in a finite number of operations using basic arithmetic operations and commonly known mathematical functions.

## 2. What is the sum of sine positive zero-crossings?

The sum of sine positive zero-crossings refers to the total number of times the sine function crosses the x-axis in a positive direction. This can also be thought of as the number of complete cycles of the sine wave in the positive x-axis direction.

## 3. Why is finding a closed form solution to the sum of sine positive zero-crossings important?

Finding a closed form solution to the sum of sine positive zero-crossings is important because it allows us to easily calculate the number of positive zero-crossings without having to manually count them. This can be useful in various applications, such as signal processing and Fourier analysis.

## 4. Is there a general formula for the closed form solution to the sum of sine positive zero-crossings?

Yes, there is a general formula for the closed form solution to the sum of sine positive zero-crossings. It is given by (2n + 1), where n is the number of complete cycles of the sine wave in the positive x-axis direction.

## 5. Can the closed form solution to the sum of sine positive zero-crossings be extended to other trigonometric functions?

Yes, the closed form solution to the sum of sine positive zero-crossings can be extended to other trigonometric functions such as cosine, tangent, and their reciprocal functions. However, the specific formula may differ for each function.

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