There is something confusing me in trigonometry. Specifically, in unit circles, when angle theta goes beyond 90 degrees, we can have triangles that have negative lengths in the unit circle, if you see what I mean. What confuses me about this is how can shapes have negative lengths? This is my first question.
My second question is that if the lengths of the triangle are negative, then why is the hypotenuse of the triangle/radius of the unit circle still positive? I get that it is positive because it is given by x^2 + y^2=1 (because of the square it must be positive). But still. Shouldn't it be negative because considering sides are negative, shouldn't there be some reflection of that in the hypotenuse? After,all it is facing the negative direction!
The trigonometric functions are defined for angles between 0 an 90° over the triangles. I assume you understood this idea.
For multiple reasons one wanted to extend the functions to other angles, so one has to come up with a creative idea.
If you would plot the triangles on the unit circle, you see that the trigonometric functions have something to do with the x any y coordiantes of the points on the unit circle.
More concrete: For all angles between 0 and 90° we have that $sin(\alpha) = y/x$, where $x$y and $y$ are the coordinates of the point on the unit circle.
Then people just generalized this rule to every point on the unit circle. For degrees above $90$ the definition has not directly something to do with the triangles, it's just an extension.
Hence we don't need any triangles with negative length