How to mimic 4/pi*ArcTan(x)+1 without trig

  • Thread starter thenewmans
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In summary, the conversation discusses the need for a basic math formula with certain properties, such as limiting y between -1 and 3 and having the points (-1, 0), (0, 1), and (1, 2) hit only once. The speaker initially presents a trigonometric solution, but then asks for help in finding a non-trig solution. The other person provides two non-trig formulas, with the second one being preferred due to its closer slope to 1 at x=0 and lack of singularity. The conversation ends with gratitude.
  • #1
thenewmans
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I need a basic math formula with the following properties:
* limit y between -1 and 3.
* (x, y) hits (-1, 0), (0, 1) and (1, 2).
* Each y value occurs only once.

I managed to do this with y=4/pi*ArcTan(x)+1. But I'd like to do this without trig. I got close with y=x*2/SQRT(1+x^2)+1. But it's not right. I keep thinking it's something simple and obvious. Can you help me?

NOTE: This is not homework. I'm graphing and analyzing some proportion data in excel for myself.
 
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  • #2
Greetings thenewmans! :smile:

Here's a non-trig solution:
[tex]y=\frac {\sqrt{16 x^2+9}-3} {2 x}+1[/tex]
 
  • #3
Perfect! Wow! Thank you!
 
  • #4
Just for fun, here's another one that is closer to what you came up with. :smile:
[tex]y = \frac {2x}{\sqrt{x^2+3}}+1[/tex]
 
  • #5
OK, just tried it and it's even better because the slope at x=0 is closer to 1 (45 degrees). Thanks again!
 
  • #6
It's also better because the first one had a singularity at x=0, even though its limit was correct. ;)
 

FAQ: How to mimic 4/pi*ArcTan(x)+1 without trig

How can I mimic 4/pi*ArcTan(x)+1 without using trigonometric functions?

To mimic 4/pi*ArcTan(x)+1 without trigonometric functions, you can use the inverse hyperbolic tangent function, also known as arctanh. This function can be expressed as ln[(1+x)/(1-x)]/2, which can be used to approximate the value of ArcTan(x). By multiplying this value by 4/pi and adding 1, you can mimic the original equation without using trigonometric functions.

Is it possible to mimic 4/pi*ArcTan(x)+1 without using any functions?

No, it is not possible to mimic 4/pi*ArcTan(x)+1 without using any functions. The equation itself contains functions, such as ArcTan(x), and cannot be expressed without using functions.

Can I use a series expansion to mimic 4/pi*ArcTan(x)+1?

Yes, you can use a series expansion to mimic 4/pi*ArcTan(x)+1. One way to do this is by using the Maclaurin series expansion for the inverse tangent function, which is given by x-x^3/3+x^5/5-x^7/7+... By plugging in the value of x into this series and then multiplying by 4/pi and adding 1, you can mimic the original equation.

Are there any other methods for mimicking 4/pi*ArcTan(x)+1 without trigonometric functions?

Yes, there are other methods for mimicking 4/pi*ArcTan(x)+1 without trigonometric functions. One approach is to use a graphing calculator or software to plot the original equation and then use a curve fitting tool to find a function or equation that closely matches the plotted points. Another method is to use numerical approximation techniques, such as Newton's method or the bisection method, to find a root of the original equation and then use that root to create an approximation of the equation.

Can I mimic 4/pi*ArcTan(x)+1 without using any specific programming languages?

Yes, you can mimic 4/pi*ArcTan(x)+1 without using any specific programming languages. The methods for mimicking the equation, such as using series expansions or numerical approximations, can be implemented in various programming languages. As long as the language supports basic mathematical operations and functions, you should be able to mimic the equation without any issues.

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