For what values of x is the inequality true?
8|sin x - (x - 34pi)| < 0.8
And I so far have this
-0.1 - 34pi < sin x - x < 0.1 - 34pi
I figured someone might be bored enough to help.
Is this a practical problem, or one just for the math? If it’s a practical problem, then the answer, to some precision, can be found using the “goal seek” or “solver” functions in excel. I get:
-107.237 < x < -106.391
(it uses an iterative solver, so the actual precision can be adjusted, but will always be finite)
As for coming up with a more elegant way to solve it, I’d have to think about it some more. Sometimes the best way to get rid of sinx - x is to rewrite sinx as the series:
x - x^3/3! + x^5/5! - x^7/7! …
at least you’d get rid of the sinx and your intial x’s would cancel out, then you would just have a polynomial to solve (though, granted, an infinite one at that) Again, it would depend upon the precision needed. I’m sure there’s a more elegant way, though…
For what values of x is the inequality true?
8|sin x - (x - 34pi)| < 0.8
If you put x = 34pi + z then you get
| sin z - z | < 0.1
and you can then use the series as edonelly suggests, so
sin z - z ~ (z^3)/6
so the answer you want is going to be
34pi +/- 0.6^(1/3)
which is pretty accurate. You could solve the next term if you really need the answer though.
Does that help?
You could use an integral transform to answer that question. Have you done Laplace Transforms, Fourier Transforms, or some such?
Hi all,
If this was a practical problem I would solve graphically by finding the roots of 8|Sin(x)-x+34Pi|-.8
I got the answer of 105.9604<x<107.6679
It is strange that is almost the symmetrical answer obtained by Endonnelly. Probably one of us got some bad signal somewhere.
In pure mathematically way I don’t think is possible to solve without using numerical methods or graphic methods, at least even Mathematica 4 also fails to obtain a solution.
Hope this helps,
Andrus
(or an odd number of bad signs!)
It was me. I had carried the sign through when trying to solve analytically, and then left it as a + when I put the parentheses badk in for excel (I guess you get what you pay for)
Using the corrected formula, it does give:
105.9604 < x < 107.6679
wow, thankyou to all that gave a reply. from guessing and checking, i knew the answer was somewhere near 106, kinda. it was one problem in review packet to kick of the AP Calculus course at my school.
If you put x = 34pi + z then you get
| sin z - z | < 0.1
im not sure where u got z.
You could use an integral transform to answer that question. Have you done Laplace Transforms, Fourier Transforms, or some such?
newp. but we might do those this year? im not sure.