In compressible flow there is a formula for an angle: $$ \tan \left(\theta \right)=\frac{2\cot \left(\beta \right)\left(M_1\sin ^2\left(\beta \right)-1\right)}{M_1\left(k+\cos \left(2\beta \right)\right)+2} $$ For a finite $\theta$ it can be shown that this formula can be written as a this power series: $$ \sin \left(\beta \right)=\sin \left(\mu \right)+\frac{k+1}{4\cos \left(\mu \right)}\tan \left(\theta \right)+.....\:+\:O\left(tan^2\left(\theta \right)\right)+... $$ Where $\mu = sin^{-1}(\frac{1}{M_1})$
So im looking for a exact way to write this power series in the form: $$ \sin(\beta) = \sum... $$ Forumla can be found here https://slidetodoc.com/shock-waves-and-expansion-waves-1-shock-wave/
I want to use it in a numerical project im currently working on.
$$\tan \left(\theta \right)=\frac{2\cot \left(\beta \right)\left(M\sin ^2\left(\beta \right)-1\right)}{M\left(k+\cos \left(2\beta \right)\right)+2}$$
$$\color{blue}{s=\sin(\beta)}\implies \tan \left(\theta \right)=\frac{2 \sqrt{1-s^2} \left(M s^2-1\right)}{s \left(M \left(k+1-2 s^2\right)+2\right)} \tag 1$$ Rewrite $(1)$ as $$\tan \left(\theta \right) \Big[s \left(M \left(k+1-2 s^2\right)+2\right)\Big]-2 \sqrt{1-s^2} \left(M s^2-1\right)=0 \tag 2$$ Now, make $$s=\sin(\beta)=\sum_{k=0}^n a_k\tan^k \left(\theta \right)+O\left(\tan^{n+1} \left(\theta \right)\right)$$ Replace in $(2)$ that you will expand as a Taylor series around $\tan \left(\theta \right)=0$.
Now cancel each coefficient of $\tan^k \left(\theta \right)$ one at the time.
Edit
To make the calculations simpler let $A=\frac 12M(k+1)$ to rewrite $(2)$ $$\tan \left(\theta \right) \Big[(A+1) s -M s^3\Big]- \sqrt{1-s^2} \left(M s^2-1\right)=0 \tag 3$$
This should give $$a_0=\frac{1}{\sqrt{M}}\qquad a_1=\frac{A}{2 \sqrt{(M-1) M}} \qquad a_2= \frac{A\Big[-4(M-1)+A (M+1) \Big]}{8 (M-1)^2 \sqrt{M}}$$ $$a_3=\frac{A\Big[8(M-1)^2- 4\left(2 M^2- M-1\right)A+(4 M+1)A^2\Big]}{16 (M-1)^{7/2} \sqrt{M} }$$ The next coefficient become to be really long and messy to be typed here.