esterificslo1

2023-02-26

What is the value of $\mathrm{cos}\left(\frac{\pi }{8}\right)$?

### Answer & Explanation

placerdeleermvau

Find the value of $\mathrm{cos}\left(\frac{\pi }{8}\right)$.
In the formula $\mathrm{cos}2A=2{\mathrm{cos}}^{2}A-1$,put $A=\frac{\pi }{8}$:
$\mathrm{cos}\left(2\left(\frac{\pi }{8}\right)\right)=2{\mathrm{cos}}^{2}\left(\frac{\pi }{8}\right)-1⇒\mathrm{cos}\left(\frac{\pi }{4}\right)=2{\mathrm{cos}}^{2}\left(\frac{\pi }{8}\right)-1⇒\frac{1}{2}=2{\mathrm{cos}}^{2}\left(\frac{\pi }{8}\right)-1\left[\because cos\frac{\pi }{4}=cos45°=\frac{1}{2}\right]⇒{\mathrm{cos}}^{2}\left(\frac{\pi }{8}\right)=\frac{1+\frac{1}{2}}{2}⇒{\mathrm{cos}}^{2}\left(\frac{\pi }{8}\right)=\frac{2+1}{22}⇒\mathrm{cos}\left(\frac{\pi }{8}\right)=\frac{2+1}{22}$
Now explain the expression above.
$\mathrm{cos}\frac{\pi }{8}=\frac{2+1}{22}×\frac{2}{2}⇒=\frac{2+2}{4}⇒=\frac{2}{+}$
Therefore, the value of $\mathrm{cos}\left(\frac{\pi }{8}\right)$ is $\frac{2}{+}$.

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