HP Forum Archive 05

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Trig functions with small angles Message #1 Posted by Jim on 20 Mar 2001, 4:24 p.m. I seem to recall that my old HP-33C had two types of trig functions: the regular trig functions and then a 2nd set of trig functions for small angles less than 1 degree. Apparently, the regular trig algorithm wouldn't return accurate values for small angles. Why aren't these found on calculators today? Are the algorithms used for trig functions always returning correct values for small angles?

Re: Trig functions with small angles Message #2 Posted by J. Lopez on 21 Mar 2001, 10:54 a.m., in response to message #1 by Jim I just bought an HP-33C and have been putting it through its paces to learn of its capabilities. I have both manuals that came with the calc but I don't see any mention in them to a different set of trigonometric functions for small angles. I'm sure I checked the manuals thoroughly (they aren't too big anyway). I do seem to remember from high school math something like trigonometric functions yielding results that are very approximate to the angle argument (provided the argument was in radians and small); i.e., f(x) ~= x | x is angle in radians. I'll check tonight the accuracy of the results returned by my HP-33C for small angles and compare these with those returned by my 49G (I'll check further with one TI calculator I have). Will post results tomorrow.

Re: Trig functions with small angles Message #3 Posted by John M. on 22 Mar 2001, 4:38 a.m., in response to message #1 by Jim If your calculator doesn't return proper values for these tiny angles, there is a trick that was often used on slide rules. For sines: express the angle in minutes (decimal notation) and divide by 3440. The result will be more exact the smaller the angle. Just try it out! Good luck, John

Re: Trig functions with small angles Message #4 Posted by J. Lopez on 22 Mar 2001, 5:11 p.m., in response to message #1 by Jim Well, last night I compared the results of calculating the sine of small angles (in radians) in my HP-33C and an HP-49G. For angles less than 8.72664626 E-2 (approx 5 degrees), the difference between the two calculators was in the order of 2 E-12, which I don't think is significant. Incidentally, for this angle, .087266 radians, the sine is .087156. These values tend to be closer together as the angle decreases. For example, when the angle is 3.49066 E-2 (approx 2 degrees), the sine is 3.48995 E-2 and when the angle is 1.74533 E-2 (approx. 1 degree), the sine is 1.74524 E-2. The results given by a TI-92 Plus calculator showed better precision (typically one more digit than the HP49G).

Re: Trig functions with small angles Message #5 Posted by Peter on 23 Mar 2001, 7:16 a.m., in response to message #4 by J. Lopez The small angle formulae state that for x being very small: sin(x)~x tan(x)~x cos(x)~1 As an astronomer we deal with very tiny angles, so I always take full advantage of these relations, thus I never have to call into question the accuracy of my calculator. I wouldn't recommend using the small angle formulae for anything larger than a few arcminutes, but for angles smaller than that it's a real time saver.

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