How to find the (arriving angles) $\alpha_b , \beta_b$ ?
If we know the values of two sides of triangle $b$, $c$ and angles between them $\alpha_a , \beta_a$ . The angles $\alpha$ and $\beta$ are representing the azimuth and elevation angles. I am not understanding how to implement the law of sines and cosines for this $3$D problem.
On
Try to evaluate sides left to right in the picture. The point where three dotted lines meet is imagined to be labelled with symbol $I.$
In the horizontal plane angle $\angle AIC$ is not a right angle.
$$ BI= b \sin \beta_{\alpha}$$ $$ AI= b \cos \beta_\alpha $$ $$ IC^2=AI^2+c^2 -2 AI. c. \cos d $$ $$ BC^2= a^2=BI^2+IC^2 $$ so $$ \beta_b = \sin^{-1}\frac{BI}{a} $$ $$ HC= BI \cot \beta_{_b} $$ Now take last step, apply SSA Sine Rule in triangle $IAC$
I’m supposing that the three dashed lines are the axes of a coordinate system? If so, let’s call your origin $\Bbb O$. If all is as I’ve said, I suppose that $A\Bbb OC$ is a right triangle with hypotenuse $c$, so that $\alpha_b=90^\circ-\alpha_a$, and you can calculate the length $\overline{\Bbb OC}$.
In addition, since $A\Bbb OB$ seems to be a right triangle with the known hypotenuse $b$ and vertex angle $\beta_a$. you can calculate the altitude $\overline{\Bbb OB}$, and with the knowledge of $\overline{\Bbb OC}$, you have the angle $\beta_b$.
No Law of Sines nor Cosines, just right triangles.