# What is the exact value of sin (45 + 30)

What is the exact value of $\mathrm{sin}\left(45+30\right)$?
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lavarcar2d2
Using the formula for the sine of the sum of two angles:
$\mathrm{sin}\left(\alpha +\beta \right)=\mathrm{sin}\alpha \mathrm{cos}\beta +\mathrm{cos}\alpha \mathrm{sin}\beta$
and assuming the angles are expressed in degrees:
$\mathrm{sin}\left({45}^{\circ }+{30}^{\circ }\right)=\mathrm{sin}{45}^{\circ }\mathrm{cos}{30}^{\circ }+\mathrm{cos}{45}^{\circ }\mathrm{sin}{30}^{\circ }$
$\mathrm{sin}\left({45}^{\circ }+{30}^{\circ }\right)=\frac{\sqrt{2}}{2}\ast \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\ast \frac{1}{2}$
$\mathrm{sin}\left({45}^{\circ }+{30}^{\circ }\right)=\frac{\sqrt{2}\left(\sqrt{3}+1\right)}{4}$