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Inverse Sine Function on a Calculator

Date: 5/14/96 at 19:21:38
From: Anonymous
Subject: Inverse Sin Function on Calculator

I am a math instructor teaching academic upgrading to adult students. 
My colleague and I are trying to answer a student's question about 
the inverse sin function on the calculator.

My theory is that the calculator is programmed with an equation of 
some sort to determine the angle (theta) given the sine value.  
However, we have been unable to figure out what this equation would 

Assume you know sine theta. Therefore you know the y coordinate of 
the terminal arm of theta on the unit circle. You also know that 
theta is the arc length, if the angle is measured in radians.

The problem therefore becomes how to determine the arc length given y 
without using a trigonometric function to do so.

Any ideas?

New Brunswick, Canada

Date: 12/11/96 at 01:05:38
From: Doctor Rob
Subject: Re: Inverse Sin Function on Calculator

There are two methods to find Arcsin(y) which can be used with 
calculators. The first is to use an infinite series, and truncate when 
enough accuracy is obtained. The appropriate series is the Maclaurin 
series for arcsine:

f(x) = f(0) + f'(0)x/1! + f''(0)x^2/2! + f'''(0)x^3/3! + ...

Here f(x) = Arcsin(x), so f(0) = 0, and f'(x) = (1 - x^2)^(-1/2). 
Thus f'(0) = 1. 
Now f''(x) = (-1/2)*(1 - x^2)^(-3/2)*(-2*x) = x*(1 - x^2)^(-3/2), 
so f''(0) = 0, and so on.  
Thus, Arcsin(x) = x + x^3/6 + ...

The second is an iterative approximation method known as the CORDIC
algorithm. It uses the formulas for the sine of the sum and difference 
of two angles. As a first approximation to theta = Arcsin(y) we can 
use y. This is very accurate for small angles, less so for large ones.  
Then we compute:
     sin(theta - y) = sin(theta) cos(y) - cos(theta) sin(y)
                    = y*cos(y) - sqrt(1 - y^2)*sin(y)
                    = z,
which we can do using trig functions and square roots, but without 
using inverse trig functions. Now we know the sine of a smaller angle, 
theta - y, namely z. We approximate the angle whose sine is z by z, 
and repeat. We get a computable value of sin(theta - y - z) = w, say. 
We continue this until theta - y - z - w - ... = epsilon is small 
enough to give us all the accuracy needed. Then the answer is 
theta = y + z + w + ... and we are done.

I hope this helps.

-Doctor Rob,  The Math Forum
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Associated Topics:
High School Calculators, Computers

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