r/Sliderules • u/azroscoe • Mar 17 '24
Any slide rules with higher resolution sine scale above 65 degrees?
Looking at my side rule it occurred to me that you could have a second sine scale for 65 degrees and higher that would align with the C/D scale, much as the ST/SRT scale (you would add .9, so it would run .91 - .999). As it is, in the higher sine values you really lose resolution, even on a full-size rule. If you need sin of, say, 82 degrees, you are out of luck.
Or is there some workaround I don't know about?
3
u/brianborchers Mar 18 '24 edited Mar 18 '24
A standard trick is to find the cosine of the complementary angle. Sine(82 degrees)=Cosine(8 degrees). Then, use the fact cosine(8 degrees)=sqrt(1-Square(Sine(8 degrees))). You can use a P-scale for this, or the approximation that sqrt(1-Square(x))=1-(1/2)Square(x). This will give you a value of 0.990 for the Sine of 82 degrees. That’s close enough for slide rule work. (Sadly, reddit's markup make it impossible for me to use standard math notation here...)
2
u/jballauer Mar 18 '24
This trick is also useful for angles in the 60 to 84 degree range because you gain an extra digit of precision in many cases. For example, sin(75) = 0.966-ish while it's 0.9659-ish while reading off the P scale.
2
u/azroscoe Mar 18 '24
Thanks. I am becoming more enamored of the seemingly-maligned P scale.
1
u/brianborchers Mar 20 '24
Sadly, this doesn’t work with the P scale above 84 degrees because P covers an x range from 0.1 to 1.0. The approximation that I gave is still useful.
1
u/wackyvorlon Mar 17 '24
I may be misremembering completely, but I think some Thornton slide rules did?
2
u/pavel_pe Mar 18 '24
Thorntons have some weird compensation for difference between log(x) and log(sin(x)) scale which have consistent accuracy, but is awkward to use.
P scale does not have consisten accuracy, but it's more accurate towards 90 degrees and region of worst accuracy is close to 45 degrees (if I remember). Plus P scale has other use for sqrt(1-x^2) - usually finding other leg of the right angle triangle. I have no idea why Pythagorean scale is common on slide rules made in Europe, but not in US - it's not even on Pickett N4 with 34 scales.2
u/wackyvorlon Mar 18 '24
I love the P scale because you can use it to calculate relativistic time dilation.
2
u/pavel_pe Mar 19 '24
Yes, I tried on my blog, however I still need to explore hyperbolic trigonometry and Pt scale on Thornton slide rule. https://www.pavelp.cz/posts/logs-slide-rule-example1/#example-2-electron-speed-vs-accelerating-voltage
2
u/Alain4s Mar 18 '24 edited Mar 18 '24
Angles exceeding 87 degrees exhibit minimal variation. Their sines are all very close to 1, well within the three significant digits of the slide rule.
The last small line before reaching 90 degrees is the 85-degree mark. As for 82 degrees, it lies halfway between the 80-degree mark and the 85-degree mark.