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Emphasis on not using the trig functions. For example, the problem would be something like find the point $\pi/3$ units counterclockwise from the point $(1,0)$ on the unit circle, without using trig functions. I feel like it must be possible, but I'm drawing a total blank on how it might be done.

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If it were possible, why would anyone ever bother with the trig functions? – MJD Apr 17 '14 at 15:50
You just gave the exact definition of the functions $\cos$ and $\sin$. So what do you accept as primitive operations ? – Yves Daoust Apr 17 '14 at 15:56
My guess is because it is much more complicated to do so? I honestly don't know, but it seems strange that there's no way to derive the values of the sine function using algebra and geometry. – user143837 Apr 17 '14 at 15:56
These are transcendent functions, meaning that they take an infinite number of algebraic operations to be computed. Do you want to know about numerical methods to compute approximations ? – Yves Daoust Apr 17 '14 at 15:58
Using a compass? Other than that, I don't think so as this the basis of trigonometry I'm afraid. – user88595 Apr 17 '14 at 16:02

For this very specific case we can do: call that point $(x,y)$. As the angle formed by the two radii at (1,0) and (x,y) at origin is $\pi/3$, then we see that the chord and these radii form an equilateral triangle. So $(x,y)$ is at distance 1 from the origin as well as $(1,0)$: so we have to find a simultaneous solution to the following two equations: $x^2+y^2=1$ and $(x-1)^2+ y^2=1$. Hence $x^2 =(x-1)^2 $. So $x=\frac12$ and $y=\frac{\sqrt3}2$.

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