If we have $s$ circle with the diameter $AB$ (with length $1$) and the center $O$, then we can approximate $\operatorname{chord} AC$ where $x$ represents the value of the $\angle AOC$ in degrees, and $t=90-\frac{x}{2}$.So formula is $1-\frac{2\left(\frac{{\pi}}{360}t\right)^{2}}{\left(\dfrac{\left(\frac{\pi}{360}\right)}{\sin\left(\frac{\pi}{360}\right)} +3.21916469998\cdot 10^{-11}\cdot \left(t^{2}+\frac{t^4}{90^2\cdot23.89}\right)\right)^{\,2t^{2}}}$
For example, if $x=60(t=60)$, we will get $0.500000092\ldots$,or $x=120,(t=30)$ we will get $0.866025407\dots$.So if $x<360$ then the value of the error cannot be greater than $4.52\cdot 10^{-7}$
Is there formula with a more precise approximate value for $\sin\frac{{\pi}}{360}x$ than mine?
and another question whether it can be simplified?
If you look at this question of mine, you will see that I tried to approximate the sine function (in radians) as $$f_p(x)=\sum_{n=1}^p a_n\big[(\pi-x)x\big]^n$$ the coefficients being computed minimizing
$$S_p=\int_0^\pi\big[\sin(x)-f_p(x)\big]^2\,dx$$ $f_4(x)$ satisfies the requirement and the coefficients are $$a_1=\frac{55440 \left(-8910720+1038960 \pi ^2-14196 \pi ^4+41 \pi ^6\right)}{\pi ^{11}}\sim\frac{3185}{10006}$$ $$a_2=-\frac{720720 \left(-15079680+1754640 \pi ^2-23616 \pi ^4+65 \pi ^6\right)}{\pi ^{13}}\sim\frac{301}{9332}$$ $$a_3=\frac{2882880 \left(-24504480+2847240 \pi ^2-37920 \pi ^4+101 \pi ^6\right)}{\pi ^{15}}\sim \frac{13}{11238}$$ $$a_4=-\frac{24504480 \left(-5765760+669240 \pi ^2-8844 \pi ^4+23 \pi ^6\right)}{\pi ^{17}}\sim\frac{1}{43158}$$ and the maximum error is $7.81\times 10^{-8}$. For these values, $S_4=3.83\times 10^{-15}$.