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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: Quicker Ex-Meridian Formula for Slide Rule or Calculator**

**From:**Frank Reed

**Date:**2016 Dec 5, 11:26 -0800

John Brown, you wrote:

"values of sqrt(1-x^2) - a trig identity"

Yes. It's from the Pythagorean connection between the sine and the cosine:

cos^{2}*θ* + sin^{2}*θ* = 1, or

sin *θ* = *sqrt*(1 - cos^{2}*θ*), or

cos *θ* = *sqrt*(1 - sin^{2}*θ*),

which means that if you know the sine of any quantity, you can get the cosine from the square root of 1 minus the square of the sine, and vice versa. This can be useful when you don't need to know the angle itself. If you've already calculated the cosine from some procedure, to get the sine, you can either use an inverse cosine to get the implied underlying angle, and then look up the sine of that angle. Or you can skip the angle itself and jump right to the sine. That's the advantage of the identity.

Frank Reed