Suppose we are doing a trig substitution and make some substition $x = a \sin \theta \equiv \sin \theta = \frac{x}{a}$ where the domain of x is $|x| \le a$
Then from the reference triangle we can conclude the following:
$$\sin \theta= \frac{x}{a}$$ $$\cos \theta = \frac{\sqrt{a^2-x^2}}{a}$$ $$\tan \theta = \frac{x}{\sqrt{a^2-x^2}}$$
However, when I graph these functions on this website: https://www.desmos.com/calculator
It immediately becomes clear to be that only the sine and tangent functions are equal on the domain $x \in [-a,a]$. The sine and cosine functions are only equal on the domain $x \in[0,a]$.
Does this mean that when we use a reference triangle in trig substitution integrals to reexpress sin as cosine we have to restrict the domain to $x \in [0, a]$ ?
