Understanding and Solving Trigonometric Functions by Hand I have heard of people who can do trigonometric functions by hand, without a calculator. I know that, for example, $\sin(\theta)=\text{opposite}/\text{hypotenuse}$, but calculators can also do $\sin(64.5)$, and I don't understand how that works. What really is a trigonometric function besides its ratio and how can you solve it by hand? I know that the ratio mentioned above can be used to create relationships between sides and angles, but what is the core definition of the function $\sin(x)$? I also know that if you graph $\sin(x)$, you get the wavy sinusoid but again, what is the function definition?
 A: Your questions

What really is a trigonometric function besides its ratio and how can you solve it by hand?

are mostly answered somewhere in the Wikipedia article Sine. The trigonometric functions are very important functions in mathematics
and its applications. There are many different ways to interpret them some of which
are in the article. For example, the article states

The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

Essentially they produced brief tables of chords of circles of a suitable large size.
Later mathematicians expanded on such tables.

Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents.

Thus, computing a trigonometric function involved looking in a suitable table with
perhaps linear interpolation between two
consecutive table entries. How the tables
themselves were produced by hand is another
matter. Over the past centuries
various efforts were made to publish tables
of trigonometric functions

The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596

These efforts culminated in the middle of the 20th century with the publication of the
Handbook of Mathematical Functions edited by Milton Abramowitz and Irene Stegun. The modern replacement for this is the DLMF

The new printed volume, the NIST Handbook of Mathematical Functions, serves a similar function as the original A&S, though it is heavily updated and extended. The online version, the NIST Digital Library of Mathematical Functions (DLMF), presents the same technical information along with extensions and innovative interactive features consistent with the new medium.

The reality is that aside from the people
who computed the trigonometric tables, no
one computes them by hand because the tables
are available, and for a few decades now, calculators and computers are used to do it.
The Wikipedia article has a section "Software
implementations" which states

Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs

and give a few details about such algorithms.
A: There are many many many ways to approximate these things with high precision! The “core definition” of $\sin(x)$ is really just the circle construction. If you want an explicit formula to calculate it, one way to do this would be through a Taylor series, a calculus technique. What you’re thinking about are different characterisations, as we call them, of the same function $\sin$: and the circle one is one such characterisation.
The Taylor series is: $$\sin(x)=\sum_{n\text{ odd }}^\infty(-1)^{(n-1)/2}\frac{x^n}{n!}=x-\frac{x^3}{6}+\frac{x^5}{120}-\cdots$$ from which one can compute sine to arbitrary precision. To be precise, that was a “Maclaurin series”. Other approximations and calculations exist of course - approximation theory and how computer scientists manage things is a very complex subject. Note that the $x$ in the above formula was in the angle measurement of radians.
The series I wrote is the Taylor series of sine centred at zero. When these series are centred at zero, they are called Maclaurin series. Consider sine of zero - it’s zero. Cosine of zero is one. The derivative of sine is cosine, and if you think about the other derivatives it’s periodic: sine-cosine-negative sine-negative cosine-sine. So the first second and third derivatives of sine at zero are $1,0,-1$ and you can do the rest. If you took the derivative of the series I wrote, all of those derivatives will equal the corresponding derivative of sine. For example, the third derivative of my series (at zero) is $-1$, (from the $x^3$ term) and notice how all the other terms disappear to zero at $x=0$! You’ll find it matches all the higher order derivatives of sine too.
