Content deleted Content added
slide rules |
|||
Line 21:
=== Graphic ===
The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the measure of the angle approaches zero, the difference between the approximation and the original function also approaches 0.
<gallery widths="
File:Small_angle_compair_odd.svg|'''Figure 1.''' A comparison of the basic [[odd function|odd]] trigonometric functions to {{mvar|θ}}. It is seen that as the angle approaches 0 the approximations become better.
File:Small_angle_compare_even.svg|'''Figure 2.''' A comparison of {{math|cos ''θ''}} to {{math|1 − {{sfrac|''θ''<sup>2</sup>|2}}}}. It is seen that as the angle approaches 0 the approximation becomes better.
Line 72:
==Error of the approximations==
[[File:Small angle compare error.svg|thumb|
Near zero, the [[relative error]] of the approximations {{tmath|\cos \theta \approx 1}}, {{tmath|\sin \theta \approx \theta}}, and {{tmath|\tan \theta \approx \theta}} is quadratic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error of these approximations shrinks by two orders of magnitude. The approximation {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} has relative error which is quartic in {{tmath|\theta}}: for each order of magnitude smaller the angle is, the relative error shrinks by four orders of magnitude.
Line 82:
* {{tmath|\sin \theta \approx \theta}} at about 0.24 radians (14.0°)
* {{tmath|\textstyle \cos \theta \approx 1 - \tfrac12\theta^2 }} at about 0.66 radians (37.9°)
==Slide-rule approximations==
[[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 trigonometric functions 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).
The right-hand end of the ST or SRT scale cannot be accurate for both arcsine(0.1) = 5.74 degrees and arctangent(0.1) = 5.71 degrees. 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 ==
| |||