I know that there can't be two right angles in a right-angled triangle. But, I have seen many proofs that sin 90° = 1.
The proofs make sense analytically, but how do we know that sin(90°) (or, for that matter, sin 0° = 0) is constant as is sin ɸ such that 0° < ɸ < 90°, i.e., in the latter, we know the ratio of opposite side to the hypotenuse is fixed for a given angle but how can this be known in the above case
I have seen the unit circle animations, but historically, before extending trigonometry to the unit circle in the Cartesian plane, we must have figured this out using the standard right triangle definition of trig functions. Please do give an explanation in the right triangle definition.
I assume by right triangle definitions of trig you mean, the rule that with length $1$ hypotenuse:
$$\sin(\theta) = A$$
Where A is the side opposite of theta.
If you use the law of sines on the right triangle you get that
$$\frac{\sin(90)}{H} = \frac{\sin(\theta)}{A}.$$
Plugging in the values we know:
$$\frac{\sin(90)}{1} = \frac{A}{A} = 1.$$
Hopefully I followed all of the necessary restrictions.