Inverse Sine Function on a CalculatorDate: 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 be. 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? Betty 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 Check out our web site! http://mathforum.org/dr.math/ |
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