Is it possible to all find trigonometric values without calculator?

Question

Except for 90,0,45,30,60,and other multiples of 5, are trigonometric values calculable without the help of a calculator?

Answer 1

the following incomplete fragment of text, which i found in an obscure corner of the internet, may be regarded as a partial answer to your question, given a particular interpretation of it.

TRIG FAMILY ROBINSON by Hedda Gabbler

a small party of partially mathematically competent individuals (from a planet of origin still quite unknown to our own astronomers) finds itself stranded on a habitable world after a spaceship accident. all equipment is destroyed on the collision at planetfall, but the party is otherwise largely unharmed, except for the loss of a few individuals. after dealing with the preliminary problems of ensuring an adequate supply of food and water, protecting their settlement from intrusion by indigenous fauna, and so forth, the group decides to endeavour to develop a 'temporary civilization' to occupy the time before they can be rescued, which, on the Captain's estimate, may take several hundred years or more.

one modular subtask is identified as the need to devise a method of calculating trigonometric functions, and a decision is taken to begin with computing a rough-and-ready table of cosines which can be used as a basis for interpolation.

unfortunately by a very unfortunate coincidence, all the personnel who knew calculus have perished in the crash. this rules out the application of Taylor's theorem - at least for the immediate future, until the missing knowledge can be reconstructed.

however this tragic deficiency is partially compensated for by a more fortunate happenstance. the spaceship's senior doctor has a little daughter named Oracula who suffers from a mild form of autism, and one of the symptoms of her condition is an ability to remember numbers to a great number of decimal places. she still remembers certain items from a short list of transcendental numbers given to her a few years previously by an uncle.

the few adults capable of challenging numerical computations are at ease with performing only three operations - addition, subtraction and squaring. they are assiduous, however, and willing to continue their calculations to a high degree of precision if required.

Oracula furnishes the group with very precise representations of two transcendental numbers which we may denote by $\alpha$ and $\pi$, where: $$ \alpha = 0.739085133... \\ \pi = 3.1415926... $$ by a further stroke of good fortune another member of the party recognizes that Oracula's $\alpha$ coincides with the initial digits of the transcendental number which is the unique fixed point of the cosine function. a rapid, if facile, application of a few theorems in hypothetico-deductive logic suggests to him that the tribe's best hope of survival lies in assuming that it is highly unlikely, though of course by no means impossible, that there could have been two transcendental numbers with the same first fifty digits amongst the hundred or so memorized by Oracula - since the list she had been given, designed explicitly for use by 7-year-olds, probably contained only transcendentals which have come to the attention of mathematicians by virtue of some easily-specified property.

the first phase of the table construction project is commissioned: to compute the set of numbers $\{a_n\}$ for $n=1$ to $3600$, where $$ a_n = 2^n\alpha \text{ mod } 2\pi $$

this preliminary task is accomplished using only addition and subtraction. luckily these values are fairly evenly distributed in the interval $[0,2\pi)$

the second phase is more of a long haul, namely to compute the set $\{\cos a_n\}$ using the recurrence relation: $$ \cos a_{n+1} = 2\cos a_n^2 -1 $$

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