# Trig Values Set At Four (Places) **URL:** \ **Category:** Factual Questions\ **Created:** [December 22, 2021, 1:27pm UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474 "2021-12-22T13:27:02Z")\ **Posts on this page:** 7\ **Page:** 2
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [December 23, 2021, 1:40am UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/21 "2021-12-23T01:40:53Z")
> [@Senegoid](#): > > Do typical algebra or trig textbooks even teach interpolation any more? Nope. And they no longer have any tables in them. Although statistics textbooks do. ---
**Author:** ![Senegoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/senegoid/32/6606_2.png) [@Senegoid](https://boards.straightdope.com/u/Senegoid)\ **Post date:** [December 23, 2021, 1:51am UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/22 "2021-12-23T01:51:14Z")
When I took trig, in 12th grade in 1968, it was a full semester (not a full year) class. The second half of the year was a class called “Math Analysis”, which I think was just their name for pre-calculus. Nobody learned calculus in high school. That old 1948 algebra book I mentioned is intended for a full-year college algebra class. It is far more thorough and in-depth than any dumbed-down modern textbook, although the choice of topics to include is not entirely the same as currently. Since that was before the days of pocket calculators (and even mainframe computers were not widely accessible), there was a lot of emphasis on numerical calculations with logarithms. There was also an entire chapter on working with approximate numbers. Nobody learns the rules for that anymore. ---
**Author:** ![Francis\_Vaughan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/francis_vaughan/32/3093_2.png) [@Francis\_Vaughan](https://boards.straightdope.com/u/Francis_Vaughan)\ **Post date:** [December 23, 2021, 10:21am UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/23 "2021-12-23T10:21:05Z")
Sensitivity to numerical error is something that all programmers need to be exposed to, and preferably formally taught. When I went through undergraduate classes there was Numerical Methods (2nd year, and compulsory), Numerical Analysis (3rd year, optional) and at Honours (4th year) level Advanced Numerical Analysis. The compulsory level course took students through machine epsilon, and sensitivity to round-off and precision right at the start. The classic example being an order 20 polynomial with roots at each integer around 0. A bit flip of the lowest precision bit of any coefficient yields a polynomial with no real roots. (I think I have remembered this OK, it was a veeery long time ago.) Sadly students for the most part disliked this subject. Too much like mathematics and not enough like having fun coding. It took most of us some hard won experience to realise just how critical to our day jobs this stuff was. These issues can come and bite you at any time. For years I coded stuff that just never involved floating point. Automata of various forms, operating systems internals, virtual machine interpreters. One used floating point numbers to do things like printout average run times, and nothing more. Then one suddenly finds oneself in the world of signal processing, potential fields analysis, forward modelling of fields, and it all comes crashing in. Need to get up to speed, and those dimly remembered lectures start to matter. The nature of sensitivity to precision is nuanced. At one end people will exploit things like the small angle approximation for sine. At the other end you will be working hard to ensure a calculation stays sane and avoid the edges and blowing up when angles get small. Things like degenerate geometries will drive you nuts. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [December 23, 2021, 3:44pm UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/24 "2021-12-23T15:44:25Z")
> [@Francis\_Vaughan](#): > > The nature of sensitivity to precision is nuanced. At one end people will exploit things like the small angle approximation for sine. At the other end you will be working hard to ensure a calculation stays sane and avoid the edges and blowing up when angles get small. Things like degenerate geometries will drive you nuts. Oftentimes, in those cases, it’s the small-angle approximations that will save you. Though you might need a higher-order small-angle approximation, if you have things like x - \sin(x). Or worse, 1\over{x-\sin(x)}. Which just goes to show that understanding is more important than knowledge. ---
**Author:** ![md-2000](https://avatars.discourse-cdn.com/v4/letter/m/9d8465/32.png) [@md-2000](https://boards.straightdope.com/u/md-2000)\ **Post date:** [December 23, 2021, 4:37pm UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/25 "2021-12-23T16:37:13Z")
> [@Francis\_Vaughan](#): > > The classic example being an order 20 polynomial with roots at each integer around 0. A bit flip of the lowest precision bit of any coefficient yields a polynomial with no real roots. (I think I have remembered this OK, it was a veeery long time ago.) Very simply, one of the first examples in Numerical Analysis, the prof solved for the intersection of two lines, plus error - ending up with, say, 0.003 +/- 0.005; then asks what this means for the first coefficient of a quadratic. Basically, the error is significant enough that we don’t even know if the parabola opens upward or downward, so a very relevant detail to consider for a solution. ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [December 23, 2021, 6:25pm UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/26 "2021-12-23T18:25:28Z")
> [@Senegoid](#): > > Do typical algebra or trig textbooks even teach interpolation any more? At the middle or high school level? If > [@Thudlow\_Boink](#): > > Nope. And they no longer have any tables in them. then it sounds like no, although > [@Senegoid](#): > > Now that everybody has calculators, does anyone even learn about interpolation anymore? Everyone should. It’s still useful sometimes. they do obviously learn it in the first year of university or whenever they take Numerical Analysis. > **[Aitken Interpolation -- from Wolfram MathWorld](https://mathworld.wolfram.com/AitkenInterpolation.html)** > > An algorithm similar to Neville's algorithm for constructing the Lagrange interpolating polynomial. Let f(x|x\_0,x\_1,...,x\_k) be the unique polynomial of kth polynomial order coinciding with f(x) at x\_0, ..., x\_k. Then f(x|x\_0,x\_1) = 1/(x\_1-x\_0)|f\_0... ---
**Author:** ![hajario](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hajario/32/171_2.png) [@hajario](https://boards.straightdope.com/u/hajario)\ **Post date:** [December 23, 2021, 7:53pm UTC](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474/27 "2021-12-23T19:53:10Z")
> [@naita](#): > > 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. This is a very clever observation and I would bet that that is the reason. [Previous page](https://boards.straightdope.com/t/trig-values-set-at-four-places/956474.md?page=1)