Finding an angle given two sides and two angles

Question

Consider the triangle given in 1. How to compute uniquely $\beta$ given $a$, $b$, $\alpha$, and $\gamma$? The double strikes on the figure indicate that the lines are parallel.

Answer 1

Using the law of cosines, we are able to find the value of the third side of the triangle, $c$:

$$c = \sqrt{a^2+b^2-2ab\cos(\alpha+\gamma)}.$$

We can use again the law of cosines to find the angle $\delta$. In particular, we start from this:

$$b^2 = a^2 + c^2 - 2ac \cos(\delta),$$

and we arrive to:

$$\delta = \arccos\left(\frac{c^2+a^2-b^2}{2ac}\right).$$

Finally, we can easily obtain $\beta$:

$$\beta = \pi - \alpha - \delta.$$