a definition of cosine that would allow for obtuse-angle cosines

Question

I've learned from here (though not fully understood) that a cosine of an obtuse angle is a notion that makes perfect sense for mathematicians.

To my highly unmathematical mind, it looks very contradictory to the basic and original definition of cosine that I learned at school: a ratio of the adjacent leg of an angle in the right triangle to the hypotenuse. According to this basic definition the cosine is something that pertains only to the right triangle, that is, a triangle in which by definition there cannot be any obtuse angles. So, continuing the logic of that definition, there cannot be such thing like a cosine of an obtuse angle. It's almost like to say "to initiate a truce during the time of peace" - since a truce is possible only in the context of war, it simply can't be initiated during the time of peace.

So, I wonder, is it possible to come up with a more extensive definition of a cosine that would be also inclusive of such cases like cosines of obtuse angles.

Take a look at https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:trig/x2ec2f6f830c9fb89:unit-circle/v/unit-circle-definition-of-trig-functions-1 — Célio Augusto, Feb 15 '23 at 03:01
For example, the unit circle definitions of sine and cosine? — peterwhy, Feb 15 '23 at 03:02
This unit circle definition is also the one given in the answer to your previous question -- the original (right-angled triangle) definition has been expanded somehow. — peterwhy, Feb 15 '23 at 03:12
@CélioAugusto - Thank you! That video has finally resolved this issue for me! — brilliant, Feb 15 '23 at 03:19
@Blue - Thank you! Very interesting. Besides the clear and vivid explanation, I also enjoyed your wording and your humor in that answer. — brilliant, Feb 15 '23 at 07:33

score 1 · Accepted Answer · answered Feb 15 '23 at 03:03

The general definition goes as follows. Take the plane $\mathbb R^2$, and a point $(x,y)$ on the unit circle centred at the origin. Now connect $(x,y)$ to $(0,0)$ with a segment line. This will draw an angle $\theta$ with the axis $x > 0, y = 0$, that you should take with a certain orientation. By definition, $\cos \theta$ is just $x$ (which can be of course negative).

a definition of cosine that would allow for obtuse-angle cosines

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