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Question Number 189463 by Rupesh123 last updated on 17/Mar/23

Answered by witcher3 last updated on 17/Mar/23

$$\cos (60+20)$$$$\cos \left(40\right)=\cos (60-20)$$$$=\frac{1}{2}(\cos \left(20\right)+\sqrt{3}\sin \left(20\right))$$$$\cos 80=\frac{1}{2}(\cos \left(20\right)-\sqrt{3}\sin \left(20\right))$$$$=({(3\cos \left(20\right)-1)}^2-{27\sin }^2\left(20\right))(6\cos \left(20\right)+1)$$$$=({36\cos }^2\left(20\right)-6\cos \left(20\right)-26)(6\cos \left(20\right)+1)$$$$\mathrm{c}=\cos 20\iff$$$$={216\mathrm{c}}^3-162\mathrm{c}-26=1$$$${216\mathrm{c}}^3-162\mathrm{c}-27=0$$$$\iff {8\mathrm{c}}^3-6\mathrm{c}-1=0...\left(\mathrm{E}\right)$$$$\cos (3.20)={4\cos }^3\left(20\right)-3\cos \left(20\right)=\frac{1}{2}$$$$\iff {8\cos }^3\left(20\right)-6\cos \left(20\right)-1=0$$$$\mathrm{heins}\mathrm{E}\mathrm{True}\iff$$$$(6\cos 80-1)(6\cos \left(40\right)-1)(6\cos \left(20\right)+1)=1$$$$\mathrm{c}$$

Commented by Rupesh123 last updated on 18/Mar/23

Excellent!

Commented by witcher3 last updated on 18/Mar/23

$$\mathrm{thank}\mathrm{You}\mathrm{God}\mathrm{bless}\mathrm{You}$$

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