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What is the most basic or fundamental definition of a trigonometric function, (say sine)? How is sine of an angle defined?

I looked up on wikipedia, and it seems that sine of an angle stems from this definition: In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

If sine is a ratio of lengths, then how can it ever be negative? I'm assuming of course that the length is always positive. Correct me if I'm mistaken.

Now please refer to this video at 8:52 on khanacademy: https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry/trig_iden_tutorial/v/trig-identities-part-2-parr-4-if-you-watch-the-proofs

Even if lengths may be negative (which I doubt), then while calculating cos(-a) in the video, why do we take the negative of opposite side but don't do anything to hypotenuse?

I'm comfortable with the unit circle definition of sine. If cos and sine are defined as the x and y coordinate as we move along a unit circle, then fine, I get it. But is the unit circle definition the most fundamental? Or is it also, derived from the ratio of lengths definition?

The reason why I'm splitting the hair to know the most basic or fundamental definition of sine is because, then I can safely rely upon it to be true in every case.

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    $\begingroup$ Related: math.stackexchange.com/questions/1176098/… $\endgroup$ – Michael Galuza Jul 23 '15 at 2:54
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    $\begingroup$ I think, "most basic" definitions is definition by sum of series ($\sum_{n=0}^\infty (-1)^n\frac{x^{2n+1}}{(2n+1)!}$) or by solution of DE $\ddot x + x = 0$. Or from complex exponent. $\endgroup$ – Michael Galuza Jul 23 '15 at 2:57
  • $\begingroup$ Well, I'll need to study differential equations to have some intuition about what you said. I'll come back to it after a couple of months. Thanks! $\endgroup$ – shobhu Jul 23 '15 at 11:35
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You are correct! For the ratio of lengths, it doesn't make sense to talk about negative values of sine and cosine, since lengths are not negative.

I would say that the unit circle is, to me, the most fundamental definition for trigonometric functions for doing plane trigonometry, by which I mean trigonometry using coordinate axes. Then, as you mention, it is pretty simple to understand negative values for these function.

However, when you're dealing with triangular trigonometry, as in trigonometry only dealing with shapes, then the ratio definition is the easiest and most sensical. Yes, you can place these shapes in the plane and use the other definition, but by just moving the shape so all of the angles are in quadrant I, you can use the ratio definition without worrying about signs since the two definitions coincide here.

For reference, see here.

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