Formula
$$\sin(\alpha+\beta)=\sin\alpha \, \cos\beta + \sin\beta \, \cos\alpha$$
Example
If $\alpha = 30\degree$ and $\beta= 45\degree$
$$\sin(30\degree+45\degree)=\sin30\degree \, \cos45\degree + \sin45\degree \, \cos30\degree =\\= \dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}\cdot \dfrac{\sqrt{3}}{2}=\dfrac{\sqrt{2}+\sqrt{6}}{4}$$
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