Saturday, January 25, 2020

TI 84 Plus CE: Testing Limits of the Arcsine Function

TI 84 Plus CE: Testing Limits of the Arcsine Function

Approximating the Arcsine

The approximation of the arcsine function is a difficult task.  In the task of approximating functions, sometimes it is helpful to determine bounds for approximation.  For example, the bounds determined by the Shafer-Fink double inequality:

For any x between 0 and 1:

3*x/(2 + √(1 - x)^2) ≤ arcsine x ≤ π*x/(2 + √(1 - x)^2)

Let L = 3*x/(2 + √(1 - x)^2)

Then π/3 * L = π*x/(2 + √(1 - x)^2)   (the upper limit)

TI-84 Plus CE Program SHAFFINK

"EWS 2020-01-05"
ClrHome
Disp "SHAFER-FINK","INEQUALITY","TI-84+ CE","0≤X≤1"
Radian
Prompt X
(3X)/(2+√(1-X²))→L
Lπ/3→U
(L+U)/2→V
sin^-1(X)→A
ClrHome
Disp "X : "+toString(X)
Disp "RESULTS SHAFER-FINK"
Disp "LOW: "+toString(L)
Disp "HIGH:"+toString(U)
Disp "AVG: "+toString(V)
Disp "ASIN:"+toString(A)

The program SHFFINK calculates the lower and upper bound, the average between the two, and for comparison, the actual arcsine of x.  Below are screen shots for x from x = 0 to x = 1, increments of 0.1.  At x = 0, the lower bound is more accurate, but as x approaches 1, the upper bound becomes more accurate. 



A Revised Upper Limit:  Gabriel Bercu

In his research article, Gabriel Bercu, Ph.D of the University of Galati (see Source below), proved that the upper limit can be improved.  The results:

( I )
arcsine x ≤ π*x/(2 + √(1 - x)^2) + (1 - π/3) * x
0 ≤ x ≤ 0.871433

( II )
arcsine x ≤ π*x/(2 + √(1 - x)^2) + (π - 4)*√(1 - x)/(2*√2) + π*(1 - x)/4
0.85068 ≤ x ≤ 1

The program BERCU is similar to SHAFFINK.  For clarity purposes, the program switches from (I) to (II) when x reaches .85068.

TI-84 Plus CE Program BERCU

"EWS 2020-01-05"
ClrHome
Disp "BERCU INEQUALITY","TI-84+ CE","0≤X≤1"
Radian
Prompt X
(3X)/(2+√(1-X²))→L
If X<.85068
Then
Lπ/3+(1-π/3)X→U
Else
Lπ/3+(π-4)√(1-X)/(2√(2))+π(1-X)/4→U
End
(L+U)/2→V
sin^-1(X)→A
ClrHome
Disp "X : "+toString(X)
Disp "RESULTS BERCU"
Disp "LOW: "+toString(L)
Disp "HIGH:"+toString(U)
Disp "AVG: "+toString(V)
Disp "ASIN:"+toString(A)

The program BERCU calculates the lower and upper bound, the average between the two, and for comparison, the actual arcsine of x.  Below are screen shots for x from x = 0 to x = 1, increments of 0.1.  At x = 0, the lower bound is more accurate, but as x approaches 1, the upper bound becomes more accurate. 



Source:

Bercu, Gabriel. (2017). Sharp Refinements for the Inverse Sine Function Related to Shafer-Fink’s Inequality. Mathematical Problems in Engineering. 2017. 1-5. 10.1155/2017/9237932. https://doi.org/10.1155/2017/9237932

Eddie


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