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I realized that following is apparently true: we had definitions of sine, cosine, etc. for angles of right triangle. Then, one suddenly draws X and Y axes, incorporates negative numbers, and sees that something is working well – incorporation of the negative sign to encode angles > 90. One then decides to forget about right triangle, be happy with negative numbers and all the beautiful (or "beautiful" when the juggling gets discovered) trigonometric equations that can be obtained this way. Am I right, this is how sine and cosine, etc. have been derived for obtuse angle?

Volker Amnu
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  • That's one way. Or, you can use the Taylor series, which converge for all $x$. – Gerry Myerson Mar 15 '16 at 11:36
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    FYI: Your description tracks pretty well with my discussion here, which is how I describe things to my students. – Blue Mar 15 '16 at 11:45
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    One can also define, in vectors' sense, $$\cos\theta=\frac{\vec u \cdot \vec v}{|\vec u||\vec v|}$$ $$\sin\theta=\frac{|\vec u \times \vec v|}{|\vec u||\vec v|}$$ where $\vec u$ and $\vec v$ are vectors and $\theta$ is the angle between them. – Mythomorphic Mar 15 '16 at 11:46
  • Try Googling "Triangular Functions" just to see what you come up with, then try Googling "Circular Functions" – John Joy Mar 15 '16 at 13:56

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