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I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like something is missing, something that ties all of these ideas together.

  • The Unit Circle. The unit circle is a way to organize all possible right triangles up to similarity. The sine and cosine can be defined as ratios of sides of these right triangles, though in practice the sine becomes the vertical component and the cosine the horizontal. What I want to know is how do I relate this beautiful diagram to the other constructions of sine and cosine.

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  • The orthonormal basis of the solution space to the differential equation: $y'' = -y$. In some sense this is as good of a definition as the unit circle.

  • Taylor Series. These grow clearly out of the differential equation above. How could I connect Taylor series to the concept of the unit circle? I think of partial Taylor series as "better and better" approximations of these function centered at zero (or maybe some place else, I guess it would not matter) -- so how is that concept linked to what $y'' = -y$ says about these equations?

Lastly, are there any other major representations I should also consider?

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    What do you mean by "Taylor Series. These grow clearly out of the differential equation above."?2011-07-18
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    $sin(z)= \frac{e^{iz}-e^{-iz}}{2i}$ for complex $z$ is the first that comes to mind. There are others, see http://en.wikipedia.org/wiki/Sine for a continued fraction expression for example. Also, I remember Apostol defining sine and cosine in his Calculus by some fundamental properties...2011-07-18
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    @Theo, if you solve the differential equation in series with the right initial conditions you get sine and cosine as the only solutions, with no initial conditions you still get series that are constant multiples of sine and cosine as solutions.2011-07-18
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    Ah, you mean that you can *compute* them by plugging in the power series ansatz. Yes, right. Thanks for clarifying!2011-07-18
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    My post here http://math.stackexchange.com/questions/1048/different-definitions-of-trigonometric-functions/1103#1103 , and my paper linked therefrom, gives a nice connection between the unit circle and the power series (for sine and cosine, and even secant and tangent). You may also be interested in this diagram: http://math.stackexchange.com/questions/392/intuitive-understanding-of-the-derivatives-of-sinx-and-cosx/1093#1093 .2011-07-19
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    @day late those spirals have me intrigued, I undertand how they could make alternating series that close in on sine and cosine, but I don't fully understand how they were constructed, what is a good reference... Or just tell me if it is short... That diagram is EXACTLY the sort of thing I was hoping for, provided I can crack the construction of the spirals. It makes power series a very geometric affair. Amazing!2011-07-19
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    @noteventhetutorknows: My note ( http://daylateanddollarshort.com/math/pdfs/sectan.pdf ) briefly discusses the sine-cosine spiral construction, which is due to Y.S. Chaikovsky, as revealed by Leo Gurin; here's the bibliographical item from my note: Leo S. Gurin, A problem, American Mathematical Monthly, 103, 1996, 683-686. (The secant-tangent zig-zag is my own original result.)2011-07-19
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    @Day Late Don as your comment seems to contain something that clearly answers at least one Noteven's questions maybe you should post it as an answer?2011-07-19

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