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I know there are many definitions of $\sin$ and $cosine$, and my favorite definitions are $\cos(x)=\text{Re}(e^{ix})$and $\sin(x)=\text{Im}(e^{ix})$, where $e^z=\sum_{n=0}^\infty \frac{z^n}{n!}$. However, I am looking for a pure geometric approach for the construction of these functions, or, at least, I am looking for an "as much geometric as possible" approach.

Every attempt to define these functions by a geometric approach that I have ever seen relies on undefined concepts such as "counterclockwise" and "clockwise".

If you could give me a nice reference, I'd be glad.

SchrodingersCat
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  • Although you seem to be primarily interested in the complex domain, you might appreciate this geometric representation of the power series for real $\theta$. – Blue Feb 28 '16 at 16:37
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    Perhaps it should be noted that the definition you give is true only if $x$ is real, whereas the sine and cosine can be defined for complex arguments. Whether there is a nice geometric definition of $\cos z$ for $z\in\mathbb C$ might be interesting. It can be defined by means of power series or differential equations, and probably in other ways as well. $\qquad$ – Michael Hardy Feb 28 '16 at 17:07
  • As an aside, $\sin x$ and $\cos x$ can be defined each as infinite continued fractions :) – Mr Pie May 13 '20 at 11:23

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Geometrically, $\sin \theta$ is defined as the ratio of length of the leg opposite an angle $\theta$ to the hypotenuse of a right triangle containing $\theta$.

$\cos \theta$ is defined similarly, except with the length of an adjacent leg.

You may want to read this for multiple definitions of these functions. For a definition of an angle, you can read this wikipedia page. No trigonometric functions are defined- only rays and intersection points.

zz20s
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  • I think the OP wants the complex definitions of sine and cosine. – Simply Beautiful Art Feb 28 '16 at 16:34
  • @SimpleArt Analytical continuations are geometrical, don't you think? ;-) – Stefan Mesken Feb 28 '16 at 16:36
  • I didn't read that in the OP's post. Would the OP mind chiming in here? – zz20s Feb 28 '16 at 16:36
  • The unit circle definition relies on concepts such as "clockwise' and "counter-clockwise". I don't know how to define these things in a pure geometric context. Also, I don't know how to define angle without defining the trigonometric functions first :) –  Feb 28 '16 at 16:38
  • I specifically meant the top of the link, which defines the functions with right triangles. Furthermore, you can define angle as the a figure formed by two rays. – zz20s Feb 28 '16 at 16:40
  • At best this won't work for things like $\sin91^\circ$ unless you say somewhat more than what you've said here. $\qquad$ – Michael Hardy Feb 28 '16 at 17:08
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    @ViniciusRodrigues : Suppose $A,B,C$ are points on the circle, and the chord (i.e. the straight line segment between two points on the curve) from $A$ to $B$ and the one from $B$ to $C$ do not both intersect and one radius (a radius is any straight line segment with one endpoint at the center of the circle and one on the circle). So every radius intersection the chord from $A$ to $B$ does not intersect the chord from $B$ to $C$. Then one can say that $B$ is between $A$ and $C$. Not suppose the chords from $A$ to $B$ and from $B$ to $C$ and from $C$ to $D$ and so on, are$,\ldots\qquad$ – Michael Hardy Feb 28 '16 at 17:15
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    $\ldots,$so chosen that $B$ is between $A$ and $C$, and $C$ between $B$ and $D$, and $D$ between $C$ and $E$, etc. --- say up to $Z$. Then you have a polygonal path from $A$ to $Z$ in one direction on the circle, which we might call "clockwise", and we could then work on showing that there's only one other direction. What is the sum of the lengths of these chords? Consider all such polygonal paths from $A$ to $Z$, with any finite number of segments in the path. If we can show that the sum of their lengths, over all possible such paths, has an upper bound, then$,\ldots\qquad$ – Michael Hardy Feb 28 '16 at 17:18
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    $\ldots,$the smallest such upper bound would be, by definition, the length of the curve from $A$ to $Z$. One can use such lengths of curves to define the measures of angles. $\qquad$ – Michael Hardy Feb 28 '16 at 17:19