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What is ${\displaystyle \arccos \left(x\right)}$? it is the angle given by the ratio of sides of length x and 1. That is, on a triangle with adjacent side length x and hypotenuse length 1 we will find the angle ${\displaystyle \arccos \left(x\right)}$.
Now what is ${\displaystyle \sin \left(y\right)}$? It is the ratio of the opposite side to the hypotenuse of a triangle with angle y in the respective part. So if we use ${\displaystyle y=\arccos \left(x\right)}$, we have that ${\displaystyle \sin \left(y\right)}$ is the ratio of the sides with degree given by a triangle whose adjacent and hypotenuse sides are length x and 1 respectively.
Since we have the adjacent and hypotenuse lengths, we can calculate the opposite sides length by the Pythagorean theorem. This means, if we use z for the opposite side, that ${\displaystyle z^2+x^2=1^2=1}$. Solving for z gives ${\displaystyle z=\sqrt{1-x^2}}$
Then ${\displaystyle \sin \left(y\right)}$ is the ratio of the opposite size, ${\displaystyle z=\sqrt{1-x^2}}$ and the hypotenuse, 1. We may now say ${\displaystyle \sin \left(\arccos \left(x\right)\right)=\sqrt{1-x^2}}$.