Genesis Gibbs

2022-10-08

Formal definition of trigonometric functions

I want to define trigonometric function (say sine) formally with the definition that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. But there is a problem with defining an angle (and the measure of an angle) without knowing trigonometric functions. Some people say that the measure of an angle is the ratio of the lenth of the arc to the lenght of the radius. But I don't know how to define an arc without trigonometric functions.

The solution may be to define sine and cosine with power series. I know this approach and it's fine, but I'm interested in classical definition.

So I came up with my own idea. Let $x\in (0,90)$ and let P be a model of Hilbert's plane euclidian geometry, $\mu $ is segments measure, $\nu $ is angles measure such that the emasure of the right angle is 90, and $\mathrm{\u25b3}abc$ is a right triangle in which $\nu (\mathrm{\angle}abc)=x$. Then $\mathrm{sin}x=\frac{\mu (ac)}{\mu (ab)}$.

This definition requires proving many theorems (for instance existance of measures and triangle with given angles) and you have to prove that the definition doesn't depnd on the choice of the model, choice of the segments measure and choice of the right triangle.

The question is: Can we define angles and sine without referring to Hilbert' theory? Maybe it's possible to define measure of angles in euclidian model ${\mathbb{R}}^{2}$. I think the key part of the definition must be additivity of the measure, as it is in Hilbert's theory.

I want to define trigonometric function (say sine) formally with the definition that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. But there is a problem with defining an angle (and the measure of an angle) without knowing trigonometric functions. Some people say that the measure of an angle is the ratio of the lenth of the arc to the lenght of the radius. But I don't know how to define an arc without trigonometric functions.

The solution may be to define sine and cosine with power series. I know this approach and it's fine, but I'm interested in classical definition.

So I came up with my own idea. Let $x\in (0,90)$ and let P be a model of Hilbert's plane euclidian geometry, $\mu $ is segments measure, $\nu $ is angles measure such that the emasure of the right angle is 90, and $\mathrm{\u25b3}abc$ is a right triangle in which $\nu (\mathrm{\angle}abc)=x$. Then $\mathrm{sin}x=\frac{\mu (ac)}{\mu (ab)}$.

This definition requires proving many theorems (for instance existance of measures and triangle with given angles) and you have to prove that the definition doesn't depnd on the choice of the model, choice of the segments measure and choice of the right triangle.

The question is: Can we define angles and sine without referring to Hilbert' theory? Maybe it's possible to define measure of angles in euclidian model ${\mathbb{R}}^{2}$. I think the key part of the definition must be additivity of the measure, as it is in Hilbert's theory.

Salma Baird

Beginner2022-10-09Added 8 answers

Step 1

To define an angle it is enough to have a notion of length of a smooth curve or a notion of area enclosed by a smooth, closed, simple curve. If two half-lines OP,OQ share their origin O, we may consider a unit circle $\mathrm{\Gamma}$ centered at O and consider the circle sector delimited by OP and OQ. Then the amplitude of $\hat{POQ}$ can be defined either in terms of the area of the previous circle sector or the length of its arc.

Step 2

On this set of amplitudes there is a natural equivalence relation given by the fact that a whole turn around a circle brings us in the exact original position.There also is a delicate point in dealing with $\hat{POQ}$ and $\hat{QOP}$ as the same angle or opposite angles, i.e. in using oriented lengths/areas or not.

To define an angle it is enough to have a notion of length of a smooth curve or a notion of area enclosed by a smooth, closed, simple curve. If two half-lines OP,OQ share their origin O, we may consider a unit circle $\mathrm{\Gamma}$ centered at O and consider the circle sector delimited by OP and OQ. Then the amplitude of $\hat{POQ}$ can be defined either in terms of the area of the previous circle sector or the length of its arc.

Step 2

On this set of amplitudes there is a natural equivalence relation given by the fact that a whole turn around a circle brings us in the exact original position.There also is a delicate point in dealing with $\hat{POQ}$ and $\hat{QOP}$ as the same angle or opposite angles, i.e. in using oriented lengths/areas or not.

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