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- 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?

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?