Several google searches in regards to this only led me to results about trigonometric substitution in regards to integrals, not to my specific question. My education board's math book and several solution sites for my book, and my school and tutors all seem to use this rather strange method for many questions which I am struggling to intuit
For example, a question like the below:
From what I understand, you take an expression that just so happens to be similar in form to a previously taught trigonometric expression, and substitute a trigonometric function in it that just so happens to work for simplification. That is where I am confused. I have asked my tutor, and a few people I know and still am struggling to understand this
It seems arbitrary. How can we say we 'simplified' the equation and got a simpler general form if what we did involved us assuming x was a very specific trigonometric function, such that substituting any other function would NOT work?
To me it is like saying $(x+3x^2) + 2$ can be simplified to $0$, because $x + 3x^2$ always evaluates to some very special function $h(x)$, which can be represented in the special form $x + 3x^2$ to evaluate to $-2$ in all cases.
This very method seems like it would work for only a specific 'branch' right? How can one produce a general solution this way? How does it logically work?
In the interval $[-\frac{1}{2},\frac{1}{2}]$ the function $\sin(\theta)$ is almost bijective. This means that if $x\in[-\frac{1}{2},\frac{1}{2}]$ then $x$ has a correspondent $\theta$ such that $x=\sin(\theta)$. Thus you should think of $\sin(\theta)$ as being another way to write $x$.
Hence it is normal to replace $x$ by this trigonometric function. In your case it just happens that it simplifies the equation.