Solving Algebraically: Finding an x Intercept of Two Functions

[jrand26]

Hi guys

What I have isn't really a 'homework problem', but I thought I'd post it here anyway. We're doing sort of an introduction to integration currently (areas between curves, u substititon etc) and one of the problems we had involved finding the x intercept of two functions first. It ended up being

x = 200sin (pi x/120)

Our instructor told us that that was very difficult to solve algebraically and required the use of 'very fancy maths'. We just solved it graphically and continued on, but I would be interested to see the solution found algebraically.

I won't bother posting my attempt, as it isn't a homework problem I don't feel the need. I hope that's alright.

 

[Mentallic]

From what I've heard being said quite a few times on this forum, there is no algebraic solution as of the present for such an equation. Whether this means there is some very complicated mathematical process or none at all (I believe the latter), then there is no way you, or many others will understand how it works.

 

[Дьявол]

And what do you need to solve here? Is pi(x) function?

 

[Office_Shredder]

The equation sin(x)=x (and many variations on it) is called a transcendental equation. What that means is that there is no algebraic solution, i.e. a solution in which only common algebraic functions such as addition, multiplication, are used. While the meaning of algebraic changes based on context, it usually includes polynomials, sine and cosine, and exponential and logarithm functions, and that's about it.

As a straight up example, I could define w(x) to be the inverse function of $$\frac{x}{200sin(\frac{ \pi x } {120})}$$ and then your solution would be x=w(1). This exact technique is used on some equations, but for obvious reasons usually isn't very instructive

 

[jrand26]

Thanks for replying.

And what do you need to solve here? Is pi(x) function?

Need to solve for x.

So are you guys are saying that this equation is 'unsolvable'?
I may have used the term 'algebraically' incorrectly. What I meant to ask is that can you rearrange for x and get the answer out (answer is 100 iirc).

 

[rock.freak667]

So are you guys are saying that this equation is 'unsolvable'?
I may have used the term 'algebraically' incorrectly. What I meant to ask is that can you rearrange for x and get the answer out (answer is 100 iirc).

It's like Office_Shredder said, there is no way you could use normal multiplication or subtraction or factorization and such things you'd normally use to solve the equation.

You could make an iterative formula such as

$$x_{n+1}=200sin(\frac{\pi x_n}{120})$$

then find a suitable interval where you think the root lies, and then choose an x₁, find x₂,x₃ and when you find the 'x' values are the same, you can take the root as that number.

 

[jrand26]

Interesting, thanks for your help.