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Question-189463




Question Number 189463 by Rupesh123 last updated on 17/Mar/23
Answered by witcher3 last updated on 17/Mar/23
cos(60+20)  cos(40)=cos(60−20)  =(1/2)(cos(20)+(√3)sin(20))  cos80=(1/2)(cos(20)−(√3)sin(20))  =((3cos(20)−1)^2 −27sin^2 (20))(6cos(20)+1)  =(36cos^2 (20)−6cos(20)−26)(6cos(20)+1)  c=cos20⇔  =216c^3 −162c−26=1  216c^3 −162c−27=0  ⇔8c^3 −6c−1=0...(E)  cos(3.20)=4cos^3 (20)−3cos(20)=(1/2)  ⇔8cos^3 (20)−6cos(20)−1=0  heins E   True⇔  (6cos80−1)(6cos(40)−1)(6cos(20)+1)=1  c
$$\mathrm{cos}\left(\mathrm{60}+\mathrm{20}\right) \\ $$$$\mathrm{cos}\left(\mathrm{40}\right)=\mathrm{cos}\left(\mathrm{60}−\mathrm{20}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\left(\mathrm{20}\right)+\sqrt{\mathrm{3}}\mathrm{sin}\left(\mathrm{20}\right)\right) \\ $$$$\mathrm{cos80}=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\left(\mathrm{20}\right)−\sqrt{\mathrm{3}}\mathrm{sin}\left(\mathrm{20}\right)\right) \\ $$$$=\left(\left(\mathrm{3cos}\left(\mathrm{20}\right)−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{27sin}^{\mathrm{2}} \left(\mathrm{20}\right)\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right) \\ $$$$=\left(\mathrm{36cos}^{\mathrm{2}} \left(\mathrm{20}\right)−\mathrm{6cos}\left(\mathrm{20}\right)−\mathrm{26}\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right) \\ $$$$\mathrm{c}=\mathrm{cos20}\Leftrightarrow \\ $$$$=\mathrm{216c}^{\mathrm{3}} −\mathrm{162c}−\mathrm{26}=\mathrm{1} \\ $$$$\mathrm{216c}^{\mathrm{3}} −\mathrm{162c}−\mathrm{27}=\mathrm{0} \\ $$$$\Leftrightarrow\mathrm{8c}^{\mathrm{3}} −\mathrm{6c}−\mathrm{1}=\mathrm{0}…\left(\mathrm{E}\right) \\ $$$$\mathrm{cos}\left(\mathrm{3}.\mathrm{20}\right)=\mathrm{4cos}^{\mathrm{3}} \left(\mathrm{20}\right)−\mathrm{3cos}\left(\mathrm{20}\right)=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Leftrightarrow\mathrm{8cos}^{\mathrm{3}} \left(\mathrm{20}\right)−\mathrm{6cos}\left(\mathrm{20}\right)−\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{heins}\:\mathrm{E}\:\:\:\mathrm{True}\Leftrightarrow \\ $$$$\left(\mathrm{6cos80}−\mathrm{1}\right)\left(\mathrm{6cos}\left(\mathrm{40}\right)−\mathrm{1}\right)\left(\mathrm{6cos}\left(\mathrm{20}\right)+\mathrm{1}\right)=\mathrm{1} \\ $$$$\mathrm{c} \\ $$
Commented by Rupesh123 last updated on 18/Mar/23
Excellent!
Commented by witcher3 last updated on 18/Mar/23
thank You God bless You
$$\mathrm{thank}\:\mathrm{You}\:\mathrm{God}\:\mathrm{bless}\:\mathrm{You} \\ $$

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