Computing sin without a Chart

Date: 02/23/98 at 09:49:03
From: Mark Ranum
Subject: converting a sin angle back to degrees

I need to be able to convert a sin angle back to degrees.  Is there a 
formula for doing this?  For instance, if I have  sin .10511, this is 
equivalent to 6 deg, 2 min.  But how do I determine that?  I found a 
chart that can do this for me if I look it up, but I need to be able 
to have the formula, so that I can do this without the chart.

Thanks.

Mark

Date: 02/23/98 at 13:04:26
From: Doctor Rob
Subject: Re: converting a sin angle back to degrees

You are asking for the angle whose sine is .10511 (or arcsin[.10511]),
expressed in degrees, if I understand you correctly.  Here is a 
formula that works for sines not equal to 1 (which you can do yourself 
in another way!):

   arcsin(x) = (180/Pi)*[x + (1/2)*x^3/3 + (1/2)*(3/4)*x^5/5 +
                           (1/2)*(3/4)*(5/6)*x^7/7 + ...].

Take as many terms as you need to get the desired accuracy.  To get 
the term involving x^n from the one involving x^(n-2) just preceding 
it, multiply by x^2*(n-2)^2/[n*(n-1)].  For x near 1, this will take 
quite a few terms.  For x near 0, this will take only a few terms.

Once you have the number of degrees in decimal form, the integer part 
(to the left of the decimal) will give you the whole number of 
degrees.  Then multiply the fractional part (to the right of the 
decimal) by 60 to get the decimal number of minutes.  The integer part 
of that will give you the whole number of minutes.  Then multiply the 
fractional part by 60 to get the decimal number of seconds.

In your example,

   arcsin(x) = (180/Pi)*[.10511 + .0001935 + .00000096 + ...],
             = 57.2957795...*0.1053045...,
             = 6.033504... degrees,
             = 6 degrees 2.0102 minutes,
             = 6 degrees 2 minutes 0.615 seconds.

If this is not what you wanted, write back and I'll try again!

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