Trig Values Set At Four (Places)

Factual Questions
1

I don’t know about the rest of you. But I wasn’t exposed to Trigonometry until my senior year in high school. Anyways, when we had it (it just happened to come up in one lesson), they explained to us in about a day. It isn’t that complicated to know.

But one thing I noticed. There are several ways to express Trigonometric values. They can be expressed as exact values. For example, I think the Sine of forty-five degrees is the square root of one half, isn’t it? And it can be expressed as a chart, in the back of our math book.

This question concerns that. The values were always expressed out to four places. Why four? I know we were in high school. So the amount of accuracy wasn’t really as important as learning the material. (And we had a really patient teacher, which helped a lot. I digress.) But why four places?

And I sometimes encounter it on the internet taken to four places, usually, I guess I can tell you, in high school level formats. Again why four? It is presumably accurate enough for something. But accurate enough for what? What about science, for example?

:slight_smile:

2

A diemaker once told me that they never needed to work to more than four decimal places, so it may have carried over from industry.

3

It’s completely arbitrary. Hewlett Packard calculators often defaulted to 4 decimal places, too. But there are plenty of uses with different choices.

One approach to small beam deflections that I used years ago failed to work right with 20 places and I had to use more.

Look for “arbitrary precision” online calculators if you’re curious. When I was working with the exponential integral function, which is not available in many computing environments, I found one by Casio that was pretty nice.

4

That’s probably a sign that you want to do some algebraic simplifications on your expression before you throw it at the computer.

Another consideration is just how big you want your tables to be. One more digit of precision means your tables are ten times as long, so they quickly become unmanageable, especially if it’s just a brief appendix at the back of a high school textbook.

And speaking as a high school teacher, another consideration, in the high school environment, is ease of grading. Even if four digits isn’t practically useful (it usually isn’t), you want enough digits in your answers that you can tell the difference between a student who’s doing it right and one who’s just a good guesser.

5

That’s only if the precise values are what is defining the resolution of your table. I doubt @Jim_B 's book had 10000 values of sin x.

I was wondering how standard four digits of resolution was and did an image search for “trig tables”, and though I didn’t get an answer for my original question, because the vast majority of results were modern “just a few exact values” tables, I did notice something in the first, four decimal places, trig table I found. It lists sin, cos and tan for each whole degree angle from 1 to 90, and sin 88 and 89 (and cos 1 and 2) are only distinguished by the fourth digit. That’s plausibly why a standard school book would include that much precision and no more.

6

Let’s look at a real table: The American Practical Navigator, 1802 edition

It appears what is actually tabulated are the values of 10+\log\sin(d) to five decimal places for each minute from 0 to 45 degrees, also \cos, \tan, \sec, \csc, \cot.

Example 2: Handbook of Mathematical Functions, 1972.

A few “exact values” in terms of radicals are printed. Besides that, there are quite a few tables, but, for example, \sin(x) and \cos x are given to 23 digits for x between 0.000 and 1.600, again for 23 digits for x from 1 to 100, and for 8 digits for x from 1 to 1000. The following table of \tan x has 8 digits, but, e.g. \tan(1.57) is given as 1255.7655915. Then we get 15-digit sines, cosines of values to tenths of a degree and (co)tangents/secants to half of a degree (up to 45 degrees, of course; no real table would continue up to 90), etc.

NB no mere four digit precision anywhere! And the degrees trig tables plus interpolation data take up less than a dozen pages.

7

This comment reminded me of a puzzle I encountered years ago.

There is a hexagonal box, lined with mirrors. A laser comes in at one of the corners and reflects off each interior face exactly once before exiting at the same corner it entered. What angle does it enter?

I solved in a very tedious way by calculating each angle of each reflection, which resulted in calculation burn compounding at each reflection. My answer was ever so slightly different to the actual answer. Turns out there is a much simpler way of solving it that doesn’t depend on modelling reflections explicitly.

I will award one internet point to the first person who can tell me what that method is.

8

I assume you can just consider a tiling by hexagons, as it were?

9

That’s exactly it.

10

Good observation. Beginning students sometimes assume that, if they round x to a certain number of decimal places, sin(x) or cos(x) (or arcsin(x) or arccos(x)) will be accurate to that many decimal places. It doesn’t work that way.

11

I remember an add, back in the Intel 386 days. It said “turn your 386 into a 486”. Reading between the lines, it seems what it did was replace the trig calculation software library so it would do 4 digit calculations instead of 8-digit (or maybe just added a trig lookup table). In the day, without math coprocessor chips, this saved a lot of calculation time and significantly sped up some video games. After all, the VGA resolution of the day maxed out at 800x600 pixels, so 4 digits of accuracy would still be overkill in most games. (Which is what the ad was aimed at)

The point the OP is aiming at, I assume, is what good were the extra digits, how often were they used? Considering they typically only used discrete angles, you are working with 2 digits of accuracy for the angle. Did anyone ever interpolate to get fractional angles? Presumably that is reasonably accurate when not approaching asymptotes. 4 digits is 1 in 10,000; that’s 1 inch in 833 feet, or about 1/4 inch in 200 feet. In asymptotic situations, a different strategy may be necessary.

I ran across a program on a numerical controller where the programmer used a linear scale for a result that actually depended on square root. Considering the process did not need much accuracy, the range of error for the anticipated range of input was close to acceptable…)

(Also I recall reading once that someone discovered an error a few digits into an entry in one of the earlier 1800’s trig tables, which apparently was repeated by many other publications. In the early days, such tables were complied by hand.)

I took a course in computational mathematics, where we delved into how errors propagate in mathematics. Most complex math is done in binary floating point, including conversion back and forth to decimal. So the accuracy of what a program will produce is limited by the type of floating point number representation used. 64 bit, for example (if deep distant memory serves) allows for 1 byte of exponent (+/-127) and 7 bytes of mantissa. Using the rule of thumb 10^3 = 2^10, your best accuracy with 56 bits is about 16 significant digits.

However, order of operation is important in math, particularly with iterative calculations - errors can blow up on repetition. One example cited was the early use of computers to calculate the wing spar needed for a new design of jet airliner in the 1960’s. The answer came out - 11 feet thick. Time to re-examine the algorithm.

For trig functions, the most efficient algorithms have been determined long ago to minimize calculations to get to high accuracy.

12

I know this is a hijack, but you spent one lesson in high school math class on trigonometry? I remember it was a year-long course (though this was forty years ago and my memory is imperfect.

13

I spent 5 minutes explaining to someone what trig functions actually were and what they were practically used for by surveyors etc. He told me that showed him what a whole semester of high school had failed to teach him. Theory is great, but how it applies to the real world is often a good path to understanding.

14

I’d probably want to spend at least a few days on SOH-CAH-TOA, just to make sure that the students got (and retained) it, but yeah, it doesn’t take long to cover the basics of trig. But then there are all of the identities, and the Law of Sines and the Law of Cosines, and hyperbolic trig functions, and polynomial approximations to trig functions, and dot and cross products of vectors, and so on. You can fill up a decent fraction of a high school math course just with topics related to trig.

It’s usually not its own entire course, though. What’s typical nowadays seems to be to cover the basics in geometry, then a bit more detail in pre-calc, with the most advanced parts in calculus.

Or then there’s the infamous function used in Quake for calculating the inverse of the square root of a number (to a low but good enough precision). You know you’re getting deep into programming when you see comments like

15

That’s pretty much what I got back in the early 2000s. Although the pre-calc course was framed as “Trigonometry and Pre-Calculus,” it was deemed necessary to introduce us to the trig functions in geometry, without calling it “Geometry and Basic Trigonometry” or anything like that.

I also remember that SOH-CAH-TOA (but not in those terms) was the one thing my grandpa (a math teacher) had taught me before I did this ACT test in seventh grade as part of the TAG* program. This was largely sufficient for the trigonometry I encountered in the math section, which was my best section. (I don’t remember my math score, but my overall was 26.)

It gets you pretty far, especially on multiple-choice tests.

*It stands for “Talented and Gifted.” So, yeah, it’s the program for “gifted” students, but it’s not really advanced classes so much as extra activities on top of your regular classes. Today I’d hate it, but back then it was fun.

16

Honestly, you can get most of the trig questions on the SAT/ACT just by knowing that the big angle is opposite the big side, that a right isosceles triangle is 45-45-90, that the sum of the angles of a triangle is 180º, and the Pythagorean Theorem. That’s what I knew the first time I took the SAT, in 7th or 8th grade, and it was enough to get me a score in the 99th percentile on the trig questions.

17

I’ve not actually encountered many other people who did the same thing. Was it also part of a program like I described? And did you happen to have a ceremony afterwards where they gave you a book explaining relativity (or maybe some similarly “smart person” topic)?

Also, I would have preferred if you’d clicked the Reply button on my post (rather than for the whole thread), even if you don’t quote me. I almost missed this response because I didn’t get a notification. (At least I’m assuming that’s why I didn’t get one.)

18

@Dewey_Finn I can’t explain early-to-mid-1980s, SE Michigan curriculum. Another thing is we never studied statistics. Or hyperbolas (though naturally parabolas). Or matrices. I don’t even know what a matrix is. And what were the four semesters of foreign language supposed to accomplish :wink: ?

19

It was definitely part of some sort of program, and most of the students at my (all-gifted) school took it, but I don’t remember any awards ceremony afterwards.

@Jim_B , you didn’t miss out on much with matrices. Every high school course I’ve ever seen with a section on matrices cuts off just before the part where they explain why they’re actually useful.

20

Do typical algebra or trig textbooks even teach interpolation any more?

I have an old algebra textbook, published in 1948, that has log tables in the back (to 5 decimal places) along with side-bar tables (called “proportional parts”) to assist with linear interpolation between tabled values.

When I took algebra myself (second year, IIRC) in 1968, we learned interpolation (or maybe when I took trig in 1969) but the tables didn’t include proportional parts, and we didn’t learn about using them.

Now that everybody has calculators, does anyone even learn about interpolation anymore? Everyone should. It’s still useful sometimes.