<s>[INST] $\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get. $\sin \alpha = \frac{3}{5} $ and $\cos \beta = -\frac{12}{13}$ . Find the values that $\cos(\alpha+\beta )$ can get.
Here $0<\alpha < \frac{\pi}{2}$ and $\frac{\pi}{2}<\beta<\pi$.
Yes I can find a value for $\cos(\alpha+\beta )$ by using $\cos(\alpha+\beta )=\cos \alpha \cos \beta - \sin \alpha \sin \beta$
But are there more values that  $\cos(\alpha+\beta )$ can get ?
 [/INST] We have two well-known Pythagorean triples:
$$3^2+4^2=5^2 ,\qquad 5^2+12^2=13^2\tag{1}$$
hence $\sin\alpha=\frac{3}{5}$ and $0\leq\alpha\leq\frac{\pi}{2}$ implies $\cos\alpha=\frac{4}{5}$, as well as $\cos\beta=-\frac{12}{13}$ and $\frac{\pi}{2}\leq\beta\leq\pi$ implies $\sin\beta=\frac{5}{13}$. So, by the cosine sum formula:
$$ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta = -\frac{48}{65}-\frac{15}{65}=\color{red}{-\frac{63}{65}.}\tag{2}$$
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