Question

Prove by using vectors that $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

Answer 1

Let $\overrightarrow{u}=(\sin \alpha ,\cos \alpha );\overrightarrow{v}=(\cos \beta ,\sin \beta )$

$\Rightarrow \overrightarrow{u}=\left[\cos \right(\frac{\pi }{2}-\alpha ),\sin (\frac{\pi }{2}-\alpha \left)\right]$

Also, $\Rightarrow |\overrightarrow{u}|=|\overrightarrow{v}|=1$

$\Rightarrow \overrightarrow{u}.\overrightarrow{v}=|\overrightarrow{u}||\overrightarrow{v}|\cos (\frac{\pi }{2}-\alpha -\beta )$

$\Rightarrow (\sin \alpha \cos \beta +\cos \alpha \sin \beta )=1\times 1\times \cos (\frac{\pi }{2}-(\alpha +\beta \left)\right)$

$\Rightarrow \sin \alpha \cos \beta +\cos \alpha \sin \beta =\sin (\alpha +\beta )$

Thus, $\Rightarrow \sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$