Gear working pressure angle

For beginner member in R&D, I always used to recommend to prove some formulars by themselves. By doing so, they can remember it better and apply it to another area better. 

Regarding gear in machine, pressure angle and correction coefficient of center distance can be obtained from the following calculation.

Note that even when gear has profile shift, base diameter doesn't change.

d′ cosα′ = db (base diameter)

​d cosα = db (base diameter)​

$$$$$$\Rightarrow\ d' \cos \alpha' = d \cos \alpha$$

$$\cos {\alpha }^{\mathrm{\prime }}=\frac{d}{d^{\mathrm{\prime }}}\mathrm{ cos}\alpha =\cos \alpha \cdot \frac{a_0}{a}$$

$$a\; :\; actual\; center\; distance$$

$$\frac{{\mathrm{<span\; style="font-family:\; arial;"><span\; style="font-size:\; small;">a0\; :\; standard\; center\; distance</span></span>}}_{}}{}$$

$$a_0 = \frac{r_1 + r_2}{2} \text{ ?}$$

Finally then I can get,

$$\therefore\ \alpha' = \cos^{-1} \left( \frac{ \cos \alpha }{ \frac{2\lambda}{z_1 + z_2} + 1 } \right)$$

This working gear pressure angle is used to set the gear specification in manufacturing.