# $\sin{\frac{\arccos(1-2x)}{2}}$ simplification without prior knowledge of the solution

I managed to, in the wild, acquire the function $$f(x) = \sin{\frac{\arccos(1-2x)}{2}}$$. I checked using Wolfram Alpha for possible simplifications, and found out that it equals (hover to show):

$$\sqrt{x}$$

I struggled to understand the reasons for this, and discovered this question asking about an equivalent problem (title provides the answer in a different form):

But this form of the question seems much more like an exercise than a problem-solving step-by-step method. How can one discover this relationship without prior knowledge of it (either from empirical graphed data or from an exercise)?

• Use round brackets rather than curly ones around a mathematical function's argument, as in my edit. Curly brackets, or braces, are for arguments of MathJaX functions such as \operatorname.
– J.G.
May 15, 2020 at 20:47

Let $$\theta:=\arccos(1-2x)$$ so $$x=\frac{1-\cos\theta}{2}=\sin^2\frac{\theta}{2}$$ and $$\sin\frac{\theta}{2}=\pm\sqrt{x}$$. But an arccosine $$\in[0,\,\pi]$$, so $$\sin\frac{\theta}{2}\ge0$$ and $$\sin\frac{\theta}{2}=\sqrt{x}$$.