Why is the hypotenuse in trig always positive regardless of the quadrant?

Solution 1:

The hypotenuse is a number that represents a distance, i.e. the norm of a vector. Distances are always positive.

That said, the figures are a bit misleading, because the green numbers have two different meanings: for the legs of the triangle, they represent the x- or y-coordinate, while for the longest side, they represent its actual length a.k.a. the hypotenuse. All green numbers should be positive if you want to interpret them as distances. I think the figures are just not mathematically sound.

Solution 2:

In many other textbooks, the trigonometric functions of angles outside the range of $0$ to $90$ degrees would be defined with the help of a unit circle, the circle of radius $1$ centered at the origin. Starting at the point $(1,0)$ on the circumference of that circle, we travel some distance counterclockwise around the circle until we reach a point. The path to that point determines an angle (where $360$ degrees equals one full trip around the circle), and the $x$ and $y$ coordinates of that point are the cosine and sine of that angle. The tangent of that angle is the slope of the line through that point and through the origin.

For some reason this textbook took a different approach; although the words "unit circle" appear in the right-hand margin of the excerpt you showed, no unit circle is shown. Instead we have triangles that are too large to relate easily to the unit circle; there is a circle on which the other end of the hypotenuse always lies, but it has radius $2.$

Despite this different way of trying to explain the trigonometric functions of large angles, the book calculates those functions in a way that is consistent with the unit-circle definition. Cosine is always positive when the direction of the given angle points to the right of the $y$ axis; the tangent is positive when the angle points into quadrants I or III, since then the line in that direction has positive slope; and so forth.

Solution 3:

The diagram is a little misleading. It uses double-headed arrows everywhere, but some of those arrows are labeled with a negative number, and others aren't.

When calculating $\sin$ and $\cos$ and the inverse functions, the sign of the $x$ and $y$ coordinates matter, which is why those parts are labeled as $-2$, $-1$, etc.

The lengths of these segments, of course, are positive. What is being labeled, though, is the displacement from the origin, which is negative if it's left on the $x$-axis, or down on the $y$-axis.

There's no real need to talk about where the hypotenuse points, since we already know where it points from the $x$ and $y$ displacements. So, it's labeled with a positive value.

I do see your confusion, though. A single-headed arrow pointing away from the origin would have reduced this confusion.