While the idea of measuring the sine of a right angle using a degenerate triangle (hypotenuse coincident with the opposite leg, and a base of zero length) may be intuitive to some, not everyone finds it satisfactory, as you can see from the other answers.
And what about the sine and cosine of angles between $90$ and $180$ degrees?
We need them in order to apply the law of sines and law of cosines to the obtuse vertex in an obtuse triangle.
Indeed, as you go on in mathematics you may find it desirable to find sines and cosines of angles greater than $180$ degrees, and even "angles" greater than $360$ degrees.
Mathematicians solve this problem in the definitions of trigonometric functions by extending the definitions of the sine and cosine functions so that they take care of input values of any size.
The extensions agree with the hypotenuse-opposite-adjacent definitions for angles that are greater than zero but less than a right angle, and they preserve useful properties such as angle-sum formulas and angle-difference formulas for input values outside that range.
Several of these extended definitions are listed in the answers to an earlier question,
How many ways are there to define sine and cosine?
My favorite extended definition for use at a level of mathematics where you are just starting to go beyond the elementary right-triangle definition is the unit-circle definition.
We observe that if you place a right triangle with unit-length hypotenuse on the Cartesian plane so that one leg is on the positive $x$-axis and one end of the hypotenuse is at the origin of coordinates, $(0,0),$
the $(x,y)$ coordinates of the other end of the hypotenuse are the cosine and sine (respectively) of the angle at the origin.
If we then take a sequence of triangles like this with increasing angles at the origin, the other end of the hypotenuse traces out points along the circle of unit radius centered at the origin.
The idea of the unit-circle definition is that you use the $(x,y)$ coordinates of these points as the definition of cosine and sine,
and to find cosine and sine of $90$ degrees or larger angles, you just keep going around the unit circle.
Once you accept this as a definition, there is nothing ambiguous or even special about the sine of $90$ degrees, except perhaps that it is unusually easy to find its value.