Finding What Number has the Sine of X

Date: 4/12/96 at 19:20:33
From: Anonymous
Subject: How to find what number has the sine of x

If it is given that the sine of y is x, is there a way to figure out
what y is, if x is given? For example, if x is -.9589242747, is it
possible to figure out what y is?

Date: 7/29/96 at 19:32:34
From: Doctor Jerry
Subject: Re: How to find what number has the sine of x

Great question!

The usual way of solving the equation sin(y) = -0.9589242747 is to use 
the inverse of the sine function, which is called arcsin.  We define 
arcsin x = y if sin y = x. Because the sine function repeats itself 
(remember that sin x = sin (x + 2pi) for all x), a convention must be 
made.  

It is assumed that -pi <= y <= pi.  Then y=arcsin(-0.9589242747) =
-1.28318530731. I got this from my calculator, set in radians. But 
this isn't the only value of y for which sin y = -.9589242747. In 
fact, all numbers of the form -1.28318530731 + 2pin for some integer 
n are solutions to your equation.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/