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Usually (in India, at least), in 10th grade, trigonometric functions are introduced as ratios of sides of a right triangle. For example, if we have a right triangle $\triangle ABC$ such that $\angle ABC = 90^\circ = \dfrac\pi2^c$ and $\angle C=\theta$. Then, $\sin\theta = \dfrac{\mathrm{perpendicular}}{\mathrm{hypotenuse}} = \dfrac{AB}{AC}$, $\cos\theta = \dfrac{\mathrm{base}}{\mathrm{hypotenuse}} = \dfrac{BC}{AC}$, etc.

We notice that here, if $f$ is a trigonometric function, then $f(\alpha)$, for some angle $\alpha$ is defined only for $\alpha\in(0^\circ,90^\circ)$.

Later on, in 11th grade (again, at least in India), the unit circle definition of trigonometric functions is taught where $\sin\beta$, for example is defined $\forall~\beta\in(-\infty,\infty)$ [all angles, basically].

I want to know why we need to extend the definition of trigonometric functions to all angles. Why can't they just be defined for acute angles? I asked this from a friend of mine and he said that this definition is used multiple times in physics. But as far as I know (and most probably), the unit circle definition was given at a time when physics was not so advanced. So, why exactly did the mathematician(s) who extended the definition do so? Was this extended definition created to solve some problem that could not be solved using the right triangle definition?

Thanks!

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    $\begingroup$ those functions are so usefull to be limited on a finite and small domain. $\endgroup$
    – L F
    Nov 19, 2020 at 17:03
  • $\begingroup$ @LuisFelipe True, but isn't it that those functions were used in these areas (where they became so significant) after this definition was given? So, why exactly was this definition given, then? I think the main thing I want to ask is "what might have been going on in the mind of the mathematician(s) who gave this definition?" $\endgroup$ Nov 19, 2020 at 17:05
  • $\begingroup$ trigonometric functions are very used in complex analysis, you will see some of them on number theory, on functions related to primes, bernoulli numbers, etc. Also you will find them in fourier series. Sin and Cos are functions easy to understand so, sen(kx), cos(kx) can form a basis for a infinite-dimensional vector spaces $\endgroup$
    – L F
    Nov 19, 2020 at 17:08
  • $\begingroup$ Some numeric approximation tehniques uses trigonometric functions, also machine learnings algorithms like SVM, PCA, etc, $\endgroup$
    – L F
    Nov 19, 2020 at 17:09
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    $\begingroup$ I added the tag "math-history" since it seems that OP's question isn't why it is interesting in general to extend trig functions to the whole real line, but more why it was first done historically $\endgroup$
    – Albert
    Nov 19, 2020 at 17:15

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Quoting Wikipedia:

Madhava (c. 1400) made early strides in the analysis of trigonometric functions and their infinite series expansions. He developed the concepts of the power series and Taylor series, and produced the power series expansions of sine, cosine, tangent, and arctangent.[24][25] Using the Taylor series approximations of sine and cosine, he produced a sine table to 12 decimal places of accuracy and a cosine table to 9 decimal places of accuracy. He also gave the power series of π and the angle, radius, diameter, and circumference of a circle in terms of trigonometric functions. His works were expanded by his followers at the Kerala School up to the 16th century.

From the power series expansion, it is natural to consider trigonometry functions as periodic functions defined on all of $\mathbb R$. I would think that this analytic point of view (as opposed to the geometric point of view where it might make more sense to limit to $0 \leq x \leq \frac{\pi}{2}$) is what you're asking about. It seems the motivation for the Taylor series expansion was a way to compute numerically trigonometric values.


Edit:

Power series expansions are particular formulas for expressing some special functions such as $\sin$, $\cos$, but also $\exp$ or $\ln$. They are useful for computing approximate values for instance, or to do other computations such as differentiating or integrating. At the beginning of calculus (with Newton and Leibniz in Europe), analysis was at first mostly concerned with such functions.

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  • $\begingroup$ I'm sorry. Being a 10th grader, I don't know much (or anything at all) about power series, Taylor series, etc. Where can I understand them to such a level that I'm capable to understand your answer? (Do I even need to know what they mean to understand your answer, though?) $\endgroup$ Nov 19, 2020 at 17:28

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