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Question is asked in title, and it's motivated by the realization that in every textbook and math class where trigonometric functions sin and cos are taught, they're just thrown at students without any motivation. Students are supposed to just take it on faith that sin and cos are important functions worthy of attention, but they're never told why.

An even greater atrocity is committed in more advanced courses where Euler's identity exp(iy)=cos(y)+isin(y) is quoted as a theorem needing proof. This in my view is completely wrong. The proper approach to the exponential map begins with the desire to find solutions to the differential equation f'=f. Dividing both sides by f and integrating yields the logarithm function, which (being increasing) has an inverse, the exponential map over the reals. This exponential map is the solution to f'=f by the inverse function theorem.

Wanting to extend the exponential map to the complex numbers, and wishing to keep the property exp(x+iy)=exp(x)exp(iy) we see that all we have to do is define exp(iy) for y real. Sin and Cos are just fancy words for the real and imaginary parts (respectively) of exp(iy). They do not have an independent existence (if they do, no author I've seen has ever explained where they came from). Put another way, Euler's identity is not a theorem. It is a definition of the left side by the right side. There is nothing to prove.

My question has to do with whether we can prove the basic properties of Re(exp(iy)) and Im(exp(iy)) using just the above-mentioned development and no power series representation. Taking the derivative of the squared modulus of exp(iy) I can show that it is a constant of 1 but I can't see a way to prove that the map y-->exp(iy) sends the real axis to anything other than the point (1,0) without power series (and hence we can't prove that there exists a smallest positive number, namely PI, such that exp(iPI)=-1). Any thoughts?

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    $\begingroup$ I don't know where you are from, but in my country the trigonometric functions sin x, cos x are taught in high school by means of the unit canonical circle and etc. Very thoroughly. About Euler's equality it is also explained at the highest high school mathematics level and more thoroughly at univeristy, and without differential equations (though it sometimes is mentioned) or power series (that comes later, too) $\endgroup$
    – DonAntonio
    Commented Apr 11, 2020 at 0:40
  • $\begingroup$ @DonAntonio I have no doubt they're taught very thoroughly. But they were pulled out of thin air, and if all that's done is define sin and cos as the x- and y-coordinates of the unit circle, then that's just another way of saying they're the real and imaginary parts of the map y-->exp(iy). There is nothing new here. It's the same show as everywhere else: the functions are just summoned by definition, without explaining the motivation for the definition (namely the desire the extend the real exponential function to the complex numbers). $\endgroup$
    – marcus66502
    Commented Apr 11, 2020 at 1:37
  • $\begingroup$ No, there is usually no one single reference to complex numbers in high school the first time they see trigonometric functions, and the trigonometric circle is introduced after the usual, basic definition of trig. functions on a straight triangle. Then the definition's extension on the trig. cricle is pretty natural and it can be done naturally with lost of geometry. Very pretty stuff, and it's explained that the definition's extension is becuase we want to make sine and cosine functions as any other function and stuff, introducing radians and all that. $\endgroup$
    – DonAntonio
    Commented Apr 11, 2020 at 10:02
  • $\begingroup$ @DonAntonio At the risk of sounding rude I have to say you're missing the point completely, which is that in high school no matter how they introduce trig functions they're just spoon feeding you what's already known, and you're supposed to (1) believe it's important, and (2) take it on faith as true. This is precisely the point of an axiomatic development: you develop something from scratch based on an inquiry. You talk about radians. Where does radian measure for angles come from?? Put another way, how do you know PI is half the arc length of a unit circle, and …. tbc $\endgroup$
    – marcus66502
    Commented Apr 11, 2020 at 23:07
  • $\begingroup$ @DonAntonio (cont'd) how exactly do you know that the map y-->exp(iy) wraps "real line" string around the unit circle, with the origin fixed to (0,1)? In fact, without using power series, how can you prove that the map y-->>exp(iy) is not just the constant map sending everything to (0,1)? $\endgroup$
    – marcus66502
    Commented Apr 11, 2020 at 23:10

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Many years ago I read an article (probably in a MAA journal) that started with the sine and cosine satisfying

$f''(x)+f(x) = 0 $ with $\sin(0) = 0, \sin'(0) = 1, \cos(0) = 1, \cos'(0) = 0 $.

From this you get $\sin'(x) = \cos(x), \cos'(x) = -\sin(x)$, and, mulyiplying by $f'(x)$, $\sin^2(x)+\cos^2(x) = 1$.

It also showed that $\sin$ and $\cos$ are periodic.

It may have also derived the addition formulas.

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  • $\begingroup$ I can prove that the modulus of exp(iy) is 1. I don't need the second order differential equation for that. I can just differentiate exp(iy) and get all the facts you mentioned, EXCEPT the periodicity. That's where I'm stuck. More precisely, I can't prove that the map y-->exp(iy) sends the real line to anything but (1,0) on the unit circle. $\endgroup$
    – marcus66502
    Commented Apr 11, 2020 at 7:32

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