# How do you find the exact value of sin 105 degrees?

How do you find the exact value of $\mathrm{sin}105$ degrees?
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Use $\mathrm{sin}105=\mathrm{sin}\left(60+45\right)=\mathrm{sin}60\mathrm{cos}45+\mathrm{cos}60\mathrm{sin}45$
$=\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{2}\right)\left(\frac{1}{\sqrt{2}}\right)=\frac{\sqrt{2}}{4}\left(\left(\sqrt{3}+1\right)=0.9656$ nearly.
Explanation:
$\mathrm{sin}45,\mathrm{cos}45$ and $\mathrm{sin}60$ are irrational.
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Talon Mcbride
We know $\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$
Hence $\mathrm{sin}{105}^{\circ }$
$=\mathrm{sin}\left({60}^{\circ }+{45}^{\circ }\right)$
$=\mathrm{sin}{60}^{\circ }\mathrm{cos}{45}^{\circ }+\mathrm{cos}{60}^{\circ }\mathrm{sin}{45}^{\circ }$
$=\frac{\sqrt{3}}{2}×\frac{1}{\sqrt{2}}+\frac{1}{2}×\frac{1}{\sqrt{2}}$
$=\frac{\sqrt{3}+1}{2\sqrt{2}}×\frac{\sqrt{2}}{\sqrt{2}}$
$=\frac{\sqrt{6}+\sqrt{2}}{4}$