Given a triangle $ABC$ with edges $a$, $b$, and $c$ and corresponding angles $\alpha$, $\beta$ and $\gamma$. Assume, the values of $a$, $b$ and $\gamma$ are given. Does there exist a direct theorem (like the law of cosines, the law of sines or the law of tangents) which describes the relationship between these parameters and $\alpha$?
Obviously, one can use the law of cosine to calculate $c$ and then apple the law of sine to solve the problem. But does there exist a specific formula for the relationship which can be applied without a step in between? Maybe, someone could give me some advice. I have already read the entire article on Wikipedia about trigonometry. Thanks in advance!
As you already know by cosine rule we can find $c$ as \begin{align} c&=\sqrt{a^2+b^2-2ab\cos\gamma} . \end{align}
However, it's not a good idea to use the sine rule to find $\alpha$, since in general, $\alpha$ could be obtuse. It's better to use known identity
\begin{align} b&=a\cos\gamma+c\cos\alpha \end{align}
to get the final expression for $\alpha$ in terms of $a,\ b$, and $\gamma$:
\begin{align} \alpha&=\arccos\left( \frac{b-a\cos\gamma}{\sqrt{a^2+b^2-2ab\cos\gamma}} \right) . \end{align}