Question Number 75193 by mathocean1 last updated on 08/Dec/19
$${it}\:{is}\:{given}\:{that}\:{cos}\frac{\pi}{\mathrm{5}}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{4}} \\ $$$${calculate}\:{the}\:{exact}\:{val}\mathrm{ue}\:{of}\: \\ $$$${cos}\frac{\mathrm{2}\pi}{\mathrm{5}}\:\:\:{and}\:\:{cos}\frac{\mathrm{3}\pi}{\mathrm{5}} \\ $$
Answered by $@ty@m123 last updated on 08/Dec/19
Commented by $@ty@m123 last updated on 08/Dec/19
$${Use}\:{formula}\:{no}.\:\mathrm{10}\:\&\mathrm{14}\:\left({b}\right) \\ $$
Commented by mathocean1 last updated on 08/Dec/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Answered by Rio Michael last updated on 08/Dec/19
![cos2x = 2cos^2 x−1 ⇒ cos((2π)/5) = cos[2((π/5))] = 2cos^2 (π/5)−1 = 2[((1 +(√5))/4)]^2 −1](Q75243.png)
$$\:{cos}\mathrm{2}{x}\:=\:\mathrm{2}{cos}^{\mathrm{2}} {x}−\mathrm{1} \\ $$$$\Rightarrow\:{cos}\frac{\mathrm{2}\pi}{\mathrm{5}}\:=\:{cos}\left[\mathrm{2}\left(\frac{\pi}{\mathrm{5}}\right)\right]\:=\:\mathrm{2}{cos}^{\mathrm{2}} \frac{\pi}{\mathrm{5}}−\mathrm{1}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2}\left[\frac{\mathrm{1}\:+\sqrt{\mathrm{5}}}{\mathrm{4}}\right]^{\mathrm{2}} −\mathrm{1} \\ $$$$ \\ $$