Small-angle approximation: Difference between revisions

Revision as of 17:51, 7 March 2025

Revision as of 18:13, 7 March 2025

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[[File:K&E Decilon slide rule left end with SRT scale.jpg | thumb | 300px | The left end of a [[Keuffel & Esser]] Deci-Lon slide rule, with a thin blue line added to show the values on the S, T, and SRT scales corresponding to sine and tangent values of 0.1 and 0.01. The S scale shows arcsine(0.1) = 5.74 degrees; the T scale shows arctangent(0.1) = 5.71 degrees; the SRT scale shows arcsine(0.01) = arctangent(0.01) = 0.01*180/pi = 0.573 degrees (to within "slide-rule accuracy").]]

Many [[slide rule]]s – especially "trig" and higher models – include an "ST" (sines and tangents) or "SRT" (sines, radians, and tangents) scale on the front or back of the slide, for computing with trigonometricsines functionsand tangents of angles smaller than about 0.1 radian (or within about 0.1 radian of <math>\pi/2</math>, that is, within 5.7 degrees of 90 degrees, in the case of cosines and cotangents).<ref>{{cite book |title=Communications Technician M 3 & 2 |date=1965 |publisher=Bureau of Naval Personnel |page=481 |url=https://books.google.com/books?id=FYB3o7iGvb8C&pg=PA481#v=onepage&q&f=false |access-date=7 March 2025}}</ref>

The right-hand end of the ST or SRT scale cannot be accurate to three decimal places for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees, so sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy". Some slide rules, such as the K&E Deci-Lon in the photo, calibrate to be accurate for radian conversion, at 5.73 degrees (off by nearly 0.4% for the tangent and 0.2% for the sine for angles around 5 degrees). Others are calibrated to 5.725 degrees, to balance the sine and tangent errors at below 0.3%. Either way, sines and tangents of angles near 5 degrees are given with somewhat worse than the usual expected "slide-rule accuracy".

== Angle sum and difference ==