"The definition of sin of and angle is opposite/hypotenuse" - well that only works for a right-angled triangle, and is the beginning of the definition of the sine function.
In order to extend the definition draw a unit circle with centre at the origin. Measure the angle counterclockwise from the positive $x$-axis and take a point $(x,y)$ on the circle. The sine of the angle (equivalent to opposite/hypotenuse in the first quadrant, with the hypotenuse made equal to $1$) is $y$ and the cosine of the angle is $x$. You can go round the circle more than once, so you can see that the functions are periodic.
This is why the functions are sometimes known as circular functions and underlies why they come in so surprisingly useful.
In case you are interested ...
In more advanced work the sine function is sometimes defined very differently, with the angle measured in radians ($2\pi$ radians $=360^{\circ}$). Then $$\sin x = \sum_{r=1}^\infty (-1)^{r-1}\frac{x^{2r-1}}{(2r-1)!}=\frac {e^{ix}-e^{-ix}}{2i}$$This can be applied in more general circumstances still, and the series is convenient because it converges rapidly and enables the sine function to be computed accurately for practical use. The first few terms are $$x-\frac {x^3}6+\frac {x^5}{120}-\frac {x^7}{5040}+\dots$$
Extending to other quadrants
If trigonometric identity formulas hold true so long as we are within the first quadrant, shouldn't we be able to extend our trig functions to all quadrants in this manner?
Mainly, you'd want the following identities.
$$\cos(\alpha+\theta)=\cos(\alpha)\cos(\theta)-\sin(\alpha)\sin(\theta)$$
$$\sin(\alpha+\theta)=\sin(\alpha)\cos(\theta)+\cos(\alpha)\sin(\theta)$$
Since we know $\sin\left(\frac\pi4\right)=\cos\left(\frac\pi4\right)=\frac{\sqrt2}2$, we can derive $\sin\left(\frac\pi2\right)=1$, $\cos\left(\frac\pi2\right)=0$, and so forth.
Similarly, we can derive what $\sin(-\theta)$ is by using $\sin(\theta-\theta),$ $\cos(\theta-\theta)$, and $\cos^2+\sin^2=1$, the Pythagorean identity. Two equations, and you can solve for $\cos(-\theta),\sin(-\theta)$ by substitution. Use Pythagorean identity for simplifying the answer.
It just happens to be that on the unit circle, the hypotenuse is by definition $1$, so...
$$\sin=\frac{\text{opp}}{\text{hyp}}=\text{opp}=y$$
$$\cos=\frac{\text{adj}}{\text{hyp}}=\text{adj}=x$$
And since both this definition and the one above derived by trig identities come out the same for $\theta>90\deg$ or $\pi$, then both are equally correct.
Best Answer
One defines the circular trigonometric functions in three stages.
Stage 1: This uses actual triangles. Then all the lengths are positive, and the ratios are of positive numbers, but angles only make sense in the interval $[0,\pi/2]$ radians. In this setting, we can place a right triangle so that one vertex is at the origin of our coordinate system, the leg it is on runs along the $x$-axis to the right angle, then the other leg runs parallel to the $y$-axis to the other vertex, and the hypotenuse is a line segment from the origin to this last vertex, lying entirely in the first quadrant. (Or, in the degenerate case where the second leg is zero, on the $x$-axis, which is not part of the first quadrant.) In this stage, we say things like "the sine of an angle is the ratio of the opposite to the hypotenuse".
Stage 2: We allow the hypotenuse to lie in any quadrant. Then we still construct right triangles ("reference triangles") in "standard position" as: put one of the vertices not at the right angle at the origin, put one leg along the $x$-axis, turn at right angles from the $x$-axis and run parallel to the $y$-axis for the other leg. This allows positive and negative values for the $x$- and $y$-coordinates of the hypotenuse and hence for the signed lengths of the first and second legs. Then we say things like "the sine of the angle is the ratio of the $y$-coordinate to the hypotenuse". (The length of the hypotenuse is always positive because it is always a length, not a coordinate.)
Stage 3: We generalize to angles less than $0$ and greater than $2\pi$ radians. This is where we get "coterminal angles". Everything still happens with right triangles, signed leg lengths, and on one copy of the coordinate plane, but we imagine that we have spun our angle around the origin many, many times in either direction. (This more accurately models real rotating systems -- a wheel doesn't rotate exactly once and then stop...)
Having made these generalizations, we are then ready to go back and apply what we have learned to non-right (degenerate, acute, and obtuse) triangles. Your language suggests you have a gap between Stage 1 and this generalization.