Solving Simple Arcsine and Arccosine Equations

 Solving Simple Arcsine and Arccosine Equations

Angle Measure

This document will focus on angle measurement in degrees. For radians and grads, please use the appropriate measurement.

90° = π/2 rad = 100 grad

180° = π rad = 200 grad

Simple Arcsine Equations

The calculator arcsine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: sin(180° - x) = sin(x), sin(x) = sin(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = sin(Θ) = sin(180° - Θ)

Base Solution 1:

n = sin(Θ)

⇒ Θ = arcsin(n)

Base Solution 2:

n = sin(180° - Θ)

arcsin(n) = 180° - Θ

⇒ Θ = 180° - arcsin(n)

Example:

0.67 = sin(Θ)

Base Solution 1: Θ = arcsin(0.67) ≈ 42.0670648025°

Base Solution 2: Θ = 180° - arcsin(0.67) ≈ 137.932935198°

Given n and α, solve for Θ:

n = sin(α + Θ)

Base Solution 1:

n = sin(α + Θ)

arcsin(n) = α + Θ

⇒ Θ = arcsin(n) – α

Base Solution 2:

n = sin(180° - (α + Θ))

n = sin(180° - α – Θ)

arcsin(n) = 180° - α – Θ

⇒ Θ = 180° - α – arcsin(n)

Example:

0.7757 = sin(Θ + 76°)

Base Solution 1: Θ = arcsin(0.7757) – 76° ≈ -25.1314549842°

Base Solution 2: Θ = 180° - 76° - arcsin(0.7757) = 104° - arcsin(0.7757) ≈ 53.131459842°

To get all the possible angles, add and subtract multiples of 360°.

Solving Simple Arccosine Equations

The calculator arccosine function gives: Domain: -1 ≤ x ≤ 1, Range: -90° ≤ Θ ≤ 90°

Note that for any angle x: cos(180° - x) = -cos(x), cos(x) = cos(x ± 360°*z) (z is an integer)

Given n, solve for Θ:

n = cos(Θ), n = cos(-Θ)

Base Solution 1:

n = cos(Θ)

⇒ Θ = arccos(n)

Base Solution 2:

n = cos(-Θ)

⇒ Θ = -arccos(n)

Example:

0.58 = cos(Θ)

Base Solution 1: Θ = arccos(0.58) ≈ 54.54945736°

Base Solution 2: Θ = -arccos(0.58) ≈ -54.54945736°

Given n and α, solve for Θ:

n = cos(α + Θ)

Base Solution 1:

n = cos(α + Θ)

arccos(n) = α + Θ

⇒ Θ = arccos(n) – α

Base Solution 2:

n = cos(-(α + Θ))

n = cos(-α – Θ)

arccos(n) = -α – Θ

-arccos(n) = α + Θ

⇒ Θ = -arccos(n) – α

Example:

0.6 = cos(35° + Θ)

Base Solution 1: Θ = arccos(0.6) – 35° ≈ 18.13012035°

Base Solution 2: Θ = -arccos(0.6) – 35° ≈ -88.13010235°

To get all the possible angles, add and subtract multiples of 360°.

I hope you find this useful and helpful,

Eddie

All original content copyright, © 2011-2026. 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.

Arcsine and Arccosine in terms of Arctangent

Arcsine and Arccosine in terms of Arctangent

Some Motivation

I recall reading about the Sinclair Scientific Programmable, a vintage scientific calculator that was introduced in 1975.  This calculator just had three trigonometric functions:  sine, cosine, and arctangent.  It is the aim of this blog to fill in the blanks.  

You can find more information here:  

http://www.vintagecalculators.com/html/scientific_prog_.html

In this blog entry, angles will have radian measure.  

For tangent, it's pretty easy use of the trig identity:

tan x = sin x ÷ cos x

Determining Arcsine 

Imagine the right triangle shown below:

Then:

sin Θ = x

Θ = arcsin x

By the Pythagorean Theorem:

t^2 + x^2 = 1

t^2 = 1 - x^2

t = √(1 - x^2)

tan Θ = x ÷ t

tan Θ = x ÷ √(1 - x^2)

Θ = arctan(x ÷ √(1 - x^2))

Then:

arcsin x = arctan(x ÷ √(1 - x^2))

Determining Arccosine

Most calculators for the arccos function have the range [0, π].  To accomplish this, we are going to use the identity

cos Θ = sin(π÷2 - Θ).

Let w = cos Θ

Then:  

w = cos Θ

w = sin(π÷2 - Θ)

arcsin w = π÷2 - Θ

Θ = π÷2 - arcsin w

Θ = π÷2 - arctan(w ÷ √(1 - w^2))

For any angle w:

arccos w = π÷2 - arctan(w ÷ √(1 - w^2))

Eddie 

All original content copyright, © 2011-2021.  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. 

Derivatives with Surprisingly Imaginary Results

Derivatives with Surprisingly Imaginary Results

Here are three derivatives of functions where complex numbers are involved with further algebraic simplification.  

d/dx √(a - x)^(1/2)

d/dx √(a- x)^(1/2)

= 1/2 ∙ (a - x)^(-1/2) ∙ -1

= -1/2 ∙ 1 ÷ (√(a - x))

Going a step further...

= -1/2 ∙ 1 ÷ (√(-1) ∙ √(x - a))

With √(-1) = i ,  1/i = -i

= i ÷ (2 ∙ √(x - a))

d/dx  arcsin(x + a)

d/dx arcsin(x + a)

= 1 ÷ √(1 - (x + a)^2)

= 1 ÷ √(1 - (x^2 + 2 ∙ a ∙ x + a^2))

= 1 ÷ √(-x^2 - 2 ∙ a ∙ x + 1 - a^2)

Factoring out -1 in the denominator: 

= 1 ÷ √((-1) ∙ (x^2 + 2 ∙ a ∙ x - 1 + a^2))

= 1 ÷ (i ∙ √(x^2 + 2 ∙ a ∙ x - 1 + a^2))

= -i ÷ √(x^2 + 2 ∙ a ∙ x - 1 + a^2)

d/dx e^(√(a - x)) 

d/dx e^(√(a - x)) 

= e^(√(a - x)) ∙ d/dx √(a - x)

= -e^(√(a - x)) ÷ (2 ∙ √(a - x))

With:  √(a - x) = i ∙ √(x - a) and e^(i ∙ Θ) = cos Θ + i ∙ sin Θ

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

= -e^(i ∙ √(x - a)) ÷ (2 ∙ i ∙ √(x - a))

=  i ∙ e^(i ∙ √(x - a)) ÷ (2 ∙ √(x - a))

= (i ∙ (cos √(x - a) + i ∙ sin √(x - a)) ÷ (2 ∙ √(x - a))

= (-sin √(x - a) + i ∙cos √(x - a)) ÷ (2 ∙ √(x - a))

Eddie

All original content copyright, © 2011-2021.  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. 

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


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.