Rebekah Hahn

2023-03-06

How to find the value of $\mathrm{sin}\left(112\frac{1}{2}\right)$ using the double or half angle formula?

trackrunner92yuy

As $\mathrm{sin}\left({90}^{\oplus }\theta \right)=\mathrm{cos}\theta$
$\mathrm{sin}\left({90}^{o}+{\left(22\frac{1}{2}\right)}^{o}\right)={\mathrm{cos}\left(22\frac{1}{2}\right)}^{o}$
Now as $2{\mathrm{cos}}^{2}A-1=\mathrm{cos}2A$, if $A={\left(22\frac{1}{2}\right)}^{o}$, $A={45}^{o}$
and $2{\mathrm{cos}}^{2}{\left(22\frac{1}{2}\right)}^{o}-1={\mathrm{cos}45}^{o}=\frac{1}{\sqrt{2}}$
or $2{\mathrm{cos}}^{2}{\left(22\frac{1}{2}\right)}^{o}=1+\frac{1}{\sqrt{2}}=\frac{\sqrt{2}+1}{\sqrt{2}}$
and ${\mathrm{cos}}^{2}{\left(22\frac{1}{2}\right)}^{o}=\frac{\sqrt{2}+1}{2\sqrt{2}}=\frac{2+\sqrt{2}}{4}$
${\mathrm{cos}\left(22\frac{1}{2}\right)}^{o}=\frac{\sqrt{2+\sqrt{2}}}{2}$
Therefore ${\mathrm{sin}\left(112\frac{1}{2}\right)}^{o}=\frac{\sqrt{2+\sqrt{2}}}{2}$

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