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Suppose your spaceship crashes onto a habitable planet. Everyone is OK but everything else is destroyed. There is no chance of being rescued. You have to rebuild modern technology from scratch. You will have to calculate angles, but you have no computer or calculator. You'll have to make a trig
table yourself. You know that the cosine of 60 is 0.5. Using the formula for finding the cosine for half the angle whose cosine is known and the formula for the cosine of added angles of known cosines, you could easily make a table of trig functions for each 15/16 of a degree. To calculate the functions of whole number degrees would take years. So why not designate 15/16 of a degree as one degree with 96 degrees to a right angle. you could use the same formulae to get half, quarter, eighth etc. degrees between each integer. In fact you could change to an octal or hexadecimal number system to further simplify matters for when you eventualy redevelop digital electronics.
Unit Circle
http://en.wikipedia.org/wiki/Unit_circle
Trig values come from ratios within a circle. These ratios are linked to the fundamental ratio of diameter & perimeter of all circles - Pi. The 'degrees' system is pretty arbitrary. [Jinbish, Apr 27 2010]
pi formula
http://www.math.hmc...files/20010.5.shtml
It is interesting that the discussion here mentions circles divided by powers-of-2, and that pi itself has a close connection with Base 16. [Vernon, Apr 27 2010]
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If you know your trig, there is no problem. Keep terms as fractions, use radians, keep surds (rather than irrational decimals). |
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{Edit: Thanks [px]. That was horrible.} |
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"If you know you're trig..."
How do I know if I'm trig? |
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//How do I know if I'm trig?// |
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Eugh! How did that happen?! I'm not trig, I'm appalled. |
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If you're Appalled, then who's Spartacus ? |
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[Ian Tindale] Jargon, noun "unintelligible talk, gibberish,"
from Old French jargon "a chattering" (of birds). |
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[Ian Tindale] can I quote you as an example of the
widespread social phenomenon whereby it is amusing to be
confused by mathematics? |
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Also, for the record, I am pretty sure that it wouldn't be that
difficult to calculate trig tables for regular degrees -
somebody did it once, I remember. And, for the CD, I
suspect that [Jinbish] speaks wisely. |
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would it be easier to do this and then convert back
to the normal cosign results? |
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The trig identities are only simple ratios that describe an
angle within a circle. It's the fraction of the circle you are
concerned with. So it really doesn't matter so much when you
use radian measures (i.e. Fractions of Pi are useful because
2*pi = 360•). |
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Now, anyone heard of Pythagoras? He and his Greek chums
did a lot of work with circles and triangles and stuff - and
they didn't gave a calculator... |
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[Max] not only amusing, but I have seen in others a certain pride in having no knowledge of mathematics, from people who would be ashamed if they had no knowledge of English literature, or philosophy. |
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Quite a lot of things - the arrangement of the branches in a pine cone, the arrangement of branches around the stem of a plant, etc. seem to be based on mathematical relationships like Phi (the "Golden Ratio"), or 2.61803399 (also known as the ratio between successive numbers in the Fibonacci series) - maths is everywhere. |
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If you're rebuilding modern technology from scratch, why not just ditch degrees entirely and go with radians from the ground up? |
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[hippo] "Intellectuals, particularly literary intellectuals, are
natural Luddites" (CP Snow) |
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//maths is everywhere//
sp. "math is everywhere", or "maths are everywhere", shirley? |
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//sp.// Don't think so; not according to Chambers.
Maths: singular noun, British colloq mathematics. N Amer equivalent math |
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On saying that, my copy of the concise OED leaves the door open to either.
maths n. Brit. colloq. mathematics [abbreviation]
mathematics n.pl. (usually treated as sing.) |
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But we know that Mr. [Tindale] likes Morden Studies... |
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You mean it's in ... The Twilight Zone ...? |
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Or should that be The Outer Limits ? |
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// The rest is history // |
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Or if you're John Cage, The Rest Is Silence. |
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//I have seen in others a certain pride in having no
knowledge of mathematics, from people who would be
ashamed if they had no knowledge of English literature, or
philosophy.// |
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I, conversely, am woefully ignorant of English literature, and
am indeed ashamed of this shortcoming. (I'm also ignorant
of philosophy, but that doesn't worry me. If they ever find
something out after nthousand years of trying, then they can
let me know and I'll catch up with it.) |
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And while we're on the subject (which we weren't), wouldn't
it make more sense to divide the circle into 2^n degrees -
either 256 or 512? Or even 1024 to get a rough alignment
with a base-10 system? |
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Not if you're a Sumerian or Babylonian. |
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Well, you could be - sitting somewhere in Sumeria 5000 years ago, looking at stuff on the Halfbakery through a freak wormhole in the space-time continuum... - but more seriously, does 60 have more unique factors than 2^n, for any n? |
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60 has 10 factors (not including 1 and itself);
512, or 2^n, where n=9, has 10 factors. |
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However, 2^n does not divide by 3 or 5 at any point. |
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All true, all true. But, the original problem was calculating
trig functions and, since it's supposedly easy to calculate
functions for half an angle, a power of 2 would make life
easier. Maybe. |
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Ah - but the trig functions are based on a right-angle triangle. You start getting square roots all over the shop: it all goes to hell in a hand-basket, thanks to Pythagoras. |
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No problem. We start with a bigger circle. |
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[Maxwell B], I thought of making a right angle 64 degrees, but that would make 1/6 and 1/12 of a circle fractional degrees. I found it pettier to make 64 the new 60. |
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