Saturday, November 20, 2021

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. 


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