TITLE: Why is the hypotenuse in trig always positive regardless of the quadrant?
QUESTION [24 upvotes]: I see an image like this:

In quadrant 2, even though the cosine is negative, because the x coordinate goes to the left, the hypothenuse is still positive. Why is this? It seems like the direction of x matters when determining if cos is positive or negative, why doesn't this hold for the hypothenuse too? Why is the hypotenuse always positive?
Why do some people talk about trig using triangles vs unit circles? Is one more correct than the other?

REPLY [3 votes]: I learned trigonometry in fourth grade of high school (it was considered an advanced topic, go figure) and, unfortunately, my textbook used the same conventions (without the fancy arrows to mark the lengths): segments parallel to the axes were to be considered positive or negative according to their “orientation”. Needless to say, several of my classmates understood very little about that definition.
Happily for them, the definition was soon dumped and one just had to remember the rigmarole

first quadrant: sine positive, cosine positive, tangent positive;
second quadrant: sine positive, cosine negative, tangent negative;
third quadrant: sine negative, cosine negative, tangent positive;
fourth quadrant: sine negative, cosine positive, tangent negative.

Only later, when discussing trigonometric inequalities, the unit circle was introduced, but just as a “graphic device”.
I don't know whether your book tells you how those triangles are built and should be thought of. You consider the ray forming the given angle with the positive $x$-axis, pick a nonzero distance (whatever you want), find the point $P$ on the ray having that distance from the origin and draw the perpendicular to the $x$-axis: in other words, you just consider the coordinates of $P$.
The distance from the origin is just a scale factor which becomes irrelevant because you always consider a ratio of two sides. You simply remove the possible negative signs, compute the ratio and then reinsert a minus sign if you removed just one. But, and here your doubt kicks in, the hypothenuse doesn't carry a sign because it is not generally parallel to one of the axes. Oh, well: what about the straight angle? Yes, there's a contradiction! How do they solve it? The $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$ and $360^\circ$ angles are special because the triangle is degenerate, so the definition must make particular cases for them.
Not the best way to define the trigonometric functions. Using $1$ as the arbitrary distance and defining cosine and sine as the coordinates of the intersection between the ray and the unit circle is much easier and avoids all doubts about which segments carry a sign and which ones don't. Then one can define the tangent as the ratio between sine and cosine. No special cases (except, of course, for the tangent of $90^\circ$ and $270^\circ$, which doesn't exist).

Maybe some rough history can help. Sine and cosine were first defined using right angles, which limited them to be acute. However the sine and cosine laws were soon discovered and they opened the way for defining sine and cosine also for obtuse angles. The sine of an obtuse angle must be positive in order that the sine law still unchanged for obtuse triangles; to the contrary, the cosine of an obtuse angle must be negative in order for the cosine law to hold unchanged. At the time, negative numbers were just considered “computational tricks”, not having a meaning of their own.
However, describing circular motion in physics made evident that angles should be considered of any value, positive or negative. And then the unit circle become an important tool. It actually doesn't matter what radius you use, as soon as the coordinates are divided by the length of the radius (which is positive by definition): just consider similar triangles. Radius one avoids computing the ratio.
It should be remarked that in ancient times, sine and cosine were not defined through a ratio leg/hypothenuse, but as the length of the suitable leg with a quite large hypothenuse, say $10\,000$ units, so decimal numbers were not needed. However, this length had the bad habit of popping out in formulas such as the addition formulas. At last, mathematicians realized that normalizing the hypothenuse to $1$ would simplify the work with the small drawback of having to use decimal numbers.
I'm not saying that the historical development was precisely along these lines: mathematics doesn't evolve in a linear way. It took long to arrive at easy definitions like the one with the unit circle. But textbook writers often think different.
The “oriented segments” approach is a mixture of the classical method with triangles outlined above and analytic geometry in order to cope with angles larger than the straight angle. Apparently, defining cosine and sine simply as coordinates of a point is considered “difficult” by the author of that textbook.
Just forget it: the hypothenuse is simply a scale factor and, as such, it doesn't carry a sign.