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Can anyone shed some light on this picture?

I am not interested in "$\sin$", "$\cos$", or the outdated trig functions. How do we guess the values of "$\tan$", "$\cot$", "$\sec$", and "$\csc$" from this picture? Is it respectively the distances $$\rho (A,E),\quad\rho (A,F),\quad\rho (O,E),\quad\rho (O,F)$$

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Why do you want to guess it? Just apply similarity ratios to different triangles and keep in mind that $OA = OD = 1$ – Kaster Feb 19 '13 at 22:19
We don't "guess the values [...] from the picture", we (well, some, like myself) actually define the values from the picture (at least in the first quadrant), and we derive identity relations using properties of right and similar triangles. For instance, $|AE|$ is $\tan\theta$ by declaration, but by comparing $\triangle OAE$ to $\triangle OCA$ we find that $|AE|/|OA| = |AC|/|CO|$ ... which is to say, $\tan / 1 = \sin/\cos$. You may find my Bloog post & PDF helpful in understanding the wealth of info in the figure:… – Blue Feb 19 '13 at 22:23
Well I declare the trig functions to be power series, @Blue. – Student Feb 19 '13 at 22:32
@George: Interestingly, the power series for four of the six trig functions can be interpreted geometrically:… (I'm still searching for the corresponding figures for cosecant and cotangent.) – Blue Feb 19 '13 at 22:40

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