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
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
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
All original content copyright, © 2011-2020. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
