(I already know the unit circle)
Why is it that the sine of an obtuse angle is the sine of its supplementary angle but the cosine of an obtuse angle is the negative of the cosine of its supplementary angle?
I can see of course on the unit circle that it is this way, but my question is:
Is it purely convention?
And if it is, what was the logic behind coming up with that specific convention as opposed to another one? Was it just practical to define the unit circle this way?
Why not have the cosine of obtuse angles be defined the same as the sine of obtuse angles? Why would that not work?
Best Answer
Those aren't definitions; they're just facts about how $\cos$ and $\sin$ behave in the second quadrant. Those features follow from how the trig functions are defined using the unit circle: If $(x,y)$ is the point on the unit circle obtained by rotating counterclockwise by $\theta$ radians from the point $(1, 0)$, then $\cos(\theta) := x$ and $\sin(\theta) := y$.
These definitions are built specifically to agree with the "SOH CAH TOA" definitions for $\theta$ between $0$ and $\pi/2$; because the hypotenuse is 1, by definition, the sine of an angle is simply equal to the opposite side, which is $y$, and the cosine of an angle is equal to the adjacent side, which is $x$.
For $\theta$ in the interval from $\pi/2$ to $\pi$ (i.e., $\theta$ an obtuse angle), $(x, y)$ is in the second quadrant, where $x<0$ and $y>0$. Reflecting across the $y$-axis corresponds to taking the supplementary angle, and this reflection negates $x$ and leaves $y$ unchanged. Hence the cosine of $\theta$ is given by the negative of the cosine of the supplementary angle to $\theta$, and the sine is equal to the sine of the supplementary angle.
Since the supplementary angle is given by $\pi-\theta$, these two facts can be summarized by the equations $$\cos(\pi-\theta) = -\cos(\theta)$$ and $$\sin(\pi-\theta)=\sin(\theta),$$ both of which are special cases of the more general angle addition and subtraction formulas.