New trigonometry is a sign of the time

physorg.com ^ | September 16, 2005

[RightWhale]

That could be used as a weapon or a shield. In fact, it looks like it has.

[MikefromOhio]

yeah...

thankfully I had some very good math teachers. I just looked through my Calc book and it's full of this Trig crap. I hope I can get it right.

[Professional Engineer]

It's a little worse for wear. It survuved the Great Depression after all.

[ScreamingFist]

Bad boys rape our young girls but violet gives willingly get some now...

LOL, darn Navy techs..... Bad Boys Rape Our Young Girls, but Violet Gives willingly for Gold or for Silver or for nothing..... What a blast from the past.....thanks.

[VeritatisSplendor]

The point is that square roots CAN be calculated by hand, and it's easy and fast. There is no way to calculate trig functions by hand that is remotely practical.

It's called the "divide and average" method. Say we want sqrt(7). The closest integer is 3, so start with 3. the average of 3 and 7/3 is 8/3. The average of 8/3 and 7/(8/3) is 127/48. This is correct to 4 decimal places already (2.6458).

If you do 1 more step and average 127/48 and 7/(127/48) you get 32257/12192 which is correct to 8 decimal places, but the high school textbooks always require at most 4-place accuracy because that's what the trig tables went up to.

When you do the math this way, you don't need a calculator.

[VeritatisSplendor]

I'm telling you it's the square of the sine so you know what it is, but that's not the way the book develops it. Use the law of cosines instead -- cos C = (a^2 + b^2 - c^2)/2ab.

So, the "spread" of angle C can be defined in terms of the "quadrance" --

S(C) = 1-((Q(a)+Q(b)-Q(c))^2)/(4Q(a)Q(b))

I think the book has an even simpler formula, but the one I just showed follows immediately from the law of cosines, and the calculations don't require tables or calculators.

[VeritatisSplendor]

Another way to think of the "spread" of an angle is the area of a unit rhombus with that angle.

[skinkinthegrass]

athaniel Bowditch, and his book is *STILL* in print, recently having had its 200th anniversary of continuous publication...Thanks, Thats' an incredible story.

[Right Winged American]

LOL, darn Navy techs.....

Yeah, that was a 2 level one for resistor color codes:

Bad = Black - 0
Boys=Brown - 1
Rape=Red - 2
Our=Orange - 3
Young=Yellow - 4
Girls=Green - 5
But=Blue - 6
Violet=Violet - 7
Gives=Gray - 8
Willingly=White - 9
Get=Gold - 20%
Some=Silver - 10%
Now=None - 0%

I'm Pretty sure - Kinda got fuzzy with the percentages ...

One thing about Navy training, when they taught it, you LEARNED ... or went back to chipping paint (G!)

[TheHound]

Yes but it is obfuscating the fact that sine tables or calculator just make it easier. Who the heck cares what the cosine value is, what people care about is the dimensions of a,b,c and the angles A,B & C. The actual Law of Cosines is: c^2 = a^2 + b^2 -2ab(cos C), thereby giving the ability to determine the length of c, a or b plus, if anyone cares, the value of cos C.

[TheHound]

One more thing. How is calculating S(C) any easier than cos (C), both formuli require addition, subtraction, multiplication and division.

[Trebics]

Reminds me of David Hestenes and Geometric Algebra for Physics...

[skinkinthegrass]

...Closer. :D

[geopyg]

"Ever try casting out 9's?"

I recall a very interesting Art Bell show where some math guy broke evrything down into one number - a nine i thought. I think it went something like:

8+7=15=87-15=72=7+2=9

12+32=44=1232-44=1188=1+1+8+8=18=1+8=9

657+36724=37381=65736724-37381=65699343=6+5+...= 45=4+5=9

YES! I believe this is it, although I thought he did it for subtraction too but I can't get it to work. So - why/how does this work?

[Poincare]

Sorry Bill Gates et al are my age and so probably had standard math instruction -

Sorry back at you. Gates and Jobs, along with others--programmers-- have spoken of the role of New Math in their development.

When do you think that New Math was taught?

Agreed that the calculator based math instruction widely taught in public schools is not good. Glad your daughter is excelling.

[supercat]

There are three kinds of people in the world. Those that understand math and those that don't. ;-)

There are 10 kinds of people in the world: those that understand binary, and those that don't.

[supercat]

Another way to think of the "spread" of an angle is the area of a unit rhombus with that angle.

Nice geometrical picture, but why wouldn't "sine" be more appropriate for that? The area of a rhombus is equal to the length of a side times the perpendicular distance to the other side. If the rhombus is sitting with the 'angle' in question at the origin and one side along the x axis, then the y coordinate of the other side will be the perpendicular distance in question. And what's the y coordinate? Seems like it should be the sine of the angle.

[gnarledmaw]

"Ever try casting out 9's?"

Yes, but it turned out that my holy water was corrupted and I was holding my symbol the wrong way...

[Condor51]

Neat pic.

I haven't seen a round slide rule in decades.

[JimWforBush]

I wonder when this will make it into surveying equipment, if ever.

Probably about the same time we convert from feet and inches to metric.

I don't see it happening soon