Sine, cosine, tangent: defined as ratios in a right triangle, but how to understand the negative values for angles outside 0-90 degrees? I can’t find an easy explanation for negative values of either/or sine, cosine and tangent when applied to angles outside $0-90^0$.
I tried to reason using the cosine law where is obvious that instead of cosine of an obtuse angle the negative cosine of the supplementary (acute) angle is used, but that is more of a redo of the cosine law than a reason for the negative value of the cosine of an obtuse angle.
I’ve tried to rely on unit circle to imagine $sine=\frac{vertical}{radius}$ while tracking a star. This led to $cosine=\frac{horizontal}{radius}$. I’m not into astronomy though. Then I looked at vertical measurement as positive when measured from ground up and negative when measured from ground under (when star falls under horizon). As for horizontal measurement, I took it negative ( as in “opposite direction”) after the observer needed to turn around to keep tracking a descending star. I did this in order to fit the unit circle.
This could explain the negative values the trigonometric functions take for angles outside $0-90^0$
Is there an easier explanation to why the trigonometric functions sometimes return negative values?
 A: In the case of the tangent function, $\tan(\theta)$ is simply the slope of the terminal side of $\theta$, when the angle is in standard position. Slopes can be negative, therefore tangent values can be negative.
As for cosines, if you believe the double-angle formula for cosine, apply it to a $60^\circ$ angle, and see what happens! With sines, use the identity for the sine of a difference to find $\sin(30^\circ - 60^\circ)$. These come out negative, so how are we supposed to save these formulas if the functions can only ever be positive?

A good definition that doesn't raise these questions is this: Place your angle $\theta$ in standard position, i.e., vertex at $(0,0)$, and initial side on the positive $x$-axis. If $(x,y)$ is a point on the terminal side of the angle, then define $r=\sqrt{x^2+y^2}$, which is always positive. Now we can define: $\sin(\theta)=\frac{y}{r}, \cos(\theta)=\frac{x}{r}, \tan(\theta)=\frac{y}{x}$.
A: That's just how they are defined -- wlog as the rectangular Cartesian coordinates of points on the unit circle corresponding to the angle. And we get consistent results when the angles are acute.
