## 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.

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

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