Why do we need to extend the definition of trigonometric ratios to all angles?

Question

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!

Answer 1

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.

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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.