Calculating trigonometry without calculator?

[Xforce]

TL;DR Summary

Trigonometry is vital for calculating angles and lengths, but unfortunately sometimes I forgot my calculator, what should I do?

Speaking of trigonometry without a calculator, I usually only memorizes the trig values of 30°, 45° and 60°. then by I can apply basic equations and applying to polygons or other geometry shapes I can get trig values for angles like 15° Or 75°. When people have enough time, people on Wikipedia even got the exact value of any degrees divisible by 3. However, constructing a 120-sided polygon takes way too long...
In some cases, for random degrees, it usually wants to get an estimated value, not an exact value. These manual calculations usually gives off a exact value... if we combine them using equations like sin(a+b)=sina*cosb+sinb*cosa and use a trial-and-error method, until we get the number of decimals we want, it will be too complicated. Despite that, estimating a surd is also hard.
So, is it possible to efficiently calculate the estimate of a trig ratio of random numbers? How does calculators do it? And how ancient people calculate them before the invention of computers?

 

[fresh_42]

So, is it possible to efficiently calculate the estimate of a trig ratio of random numbers?

One way is to use their Taylor expansions.

How does calculators do it?

IIRC they use certain algorithms which are more efficient than Taylor, but I have forgotten which.

And how ancient people calculate them before the invention of computers?

They didn't. They only had certain values: the ratios in a right triangle with integer side lengths. Euclid's geometry is done by ruler and compass, not by numeric.

 

[mfb]

That depends on what you call "efficiently". The simplest concept is a Taylor series but it isn't very efficient in terms of number of calculations. A standard computer algorithm is CORDIC but that relies on tables of some pre-computed values stored somewhere.

 

[FactChecker]

Without a computer, there are half-angle formulas and angle sum formulas that can be used to get values. Early mathematicians used to hire idiot savants who could generate tables. The savants could not describe how they were doing the calculations in any way that a normal person could understand.

 

[symbolipoint]

SLIDERULE, if you know how and if you have one

 

[Mark44]

IIRC they use certain algorithms which are more efficient than Taylor, but I have forgotten which.

I believe they are called CORDIC (for COordinate Rotation DIgital Computer), aka Volder's algorithm. See https://en.wikipedia.org/wiki/CORDIC.

 

[DEvens]

When computers were still many-millions of $, there were tables. You looked up the value closest to what you wanted in the table. If you needed more accuracy you used various standard formulas and interpolated. For certain special values or special ranges you used other formulas. Such as for very small angles and the sin() or tan() functions, or angles close to pi/2 for the cos() function.

The same general process was used for logs. There are entire books of similar tables, various other special functions as well such as Bessel functions and so on.

In public school, mumble years ago, when personal computer power simply was not available, I started in memorizing the log tables. I had read a science fiction story about a "boy scout on the moon" or some such. He had managed to solve his problem in his head by using memorized log tables. If you recall that you can multiply two numbers by taking the log of each, adding them, then getting the inverse-log, a log table in your head let's you do a lot of fairly complicated arithmetic. Sadly, my memory was not really up to the task.

Which reminds me of a joke. The forest ranger noticed that a certain kind of snake in his park was not successfully breeding. The tourists visiting the park were disturbing them at just the wrong moment, and nobody was happy about that, since the snakes were likely to be aggressive in such situations. So he invented a particular type of picnic bench-and-table that would allow the tourists to sit on it, and the snakes could safely hide in a compartment under the table and be undisturbed by the tourists. And an attractive feature of these picnic surfaces was they could be easily constructed from local materials. Thus leading to the moral of the story: log tables were invented so that adders could multiply.

 

[PeroK]

In my first year at secondary school we had Miss Rankine for maths. One of the few things I can remember from that class is that the part of the logarithm after the decimal point is called the "mantissa".

 

[Tom.G]

Here a a few pages with some shortcuts, probably most useful if used together.

http://www.math-shortcut-tricks.com/trigonometry-function-formulas/
http://sites.psu.edu/mmagabo/wp-content/uploads/sites/18075/2015/05/Trigonometry-Hand-Trick.pdf
above found with: https://www.google.com/search?&q=trig+function+shortcuts

Cheers,
Tom