Trigonometry

[man111]

Evaluation of [tex]\displaystyle \tan \frac{3\pi}{7}-4\sin \frac{\pi}{7}[/tex]

[S.B.]

Evaluation of [tex]\displaystyle \tan \frac{3\pi}{7}-4\sin \frac{\pi}{7}[/tex]

Честно казано не владея чужди езици и не ми е ясно точно какво се иска в тази задача.Прилича ми на израз,който трябва да се опрости...
[tex]tg\frac{3\pi}{7} - 4.sin\frac{\pi}{7} =[/tex]
[tex]= tg\displaystyle\frac{3\pi}{7} - \displaystyle\frac{4sin\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{\pi}{7}}{cos\displaystyle\frac{\pi}{7}} = tg\displaystyle\frac{3\pi}{7} - \displaystyle\frac{2sin\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}} = tg\displaystyle\frac{3\pi}{7} - \displaystyle\frac{2sin\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} =[/tex]

[tex]= tg\displaystyle\frac{3\pi}{7} - \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} = \displaystyle\frac{sin\displaystyle\frac{3\pi}{7}}{cos\displaystyle\frac{3\pi}{7}} - \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} = \displaystyle\frac{sin(\displaystyle\frac{7\pi}{7}- \displaystyle\frac{4\pi}{7})}{cos\displaystyle\frac{3\pi}{7}} - \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} =[/tex]

[tex]= \displaystyle\frac{sin(\pi - \displaystyle\frac{4\pi}{7})}{cos\displaystyle\frac{3\pi}{7}} - \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} = \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{3\pi}{7}} - \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}} =[/tex]

[tex]= sin\displaystyle\frac{4\pi}{7}(\displaystyle\frac{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7} - cos\displaystyle\frac{3\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} ) = sin\displaystyle\frac{4\pi}{7}(\displaystyle\frac{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7} - cos(\displaystyle\frac{\pi}{7} + \displaystyle\frac{2\pi}{7})}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}}) =[/tex]

[tex]= sin\displaystyle\frac{4\pi}{7}.\displaystyle\frac{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7} - cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7} + sin\displaystyle\frac{\pi}{7}.sin\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} =[/tex]

[tex]= \displaystyle\frac{sin\displaystyle\frac{4\pi}{7}.sin\displaystyle\frac{\pi}{7}.sin\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} = \displaystyle\frac{sin(\displaystyle\frac{7\pi}{7} - \displaystyle\frac{3\pi}{7})sin\displaystyle\frac{\pi}{7}.sin\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} = \displaystyle\frac{sin(\pi - \displaystyle\frac{3\pi}{7}).sin\displaystyle\frac{\pi}{7}.sin\displaystyle\frac{2\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} =[/tex]

[tex]= \displaystyle\frac{sin\displaystyle\frac{\pi}{7}.sin\displaystyle\frac{2\pi}{7}.sin\displaystyle\frac{3\pi}{7}}{cos\displaystyle\frac{\pi}{7}.cos\displaystyle\frac{2\pi}{7}.cos\displaystyle\frac{3\pi}{7}} =[/tex][tex]= tg\displaystyle\frac{\pi}{7}.tg\displaystyle\frac{2\pi}{7}.tg\displaystyle\frac{3\pi}{7}[/tex]

[vezni]

От казаното в предишния пост [tex]\tan\frac{3\pi}{7}-4\sin\frac{\pi}{7}=\frac{\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}}{\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}}[/tex]
Изчисляваме знаменателя: [tex]\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7} = -\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}=[/tex]
[tex]=-\frac{8\sin\frac{\pi}{7}\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}}{8\sin\frac{\pi}{7}}=-\frac{\sin\frac{8\pi}{7}}{8\sin\frac{\pi}{7}}=\frac{\sin\frac{\pi}{7}}{8\sin\frac{\pi}{7}}=\frac{1}{8}[/tex]

После числителя: [tex]A=\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}=\sin\frac{2\pi}{7}\sin\frac{4\pi}{7}\sin\frac{6\pi}{7}[/tex]
Полагаме [tex]z=e^{i\frac{2\pi}{7}} \Rightarrow \sin\frac{2\pi}{7}=\frac{z-z^{-1}}{2i},\sin\frac{4\pi}{7}=\frac{z^2-z^{-2}}{2i},\sin\frac{6\pi}{7}=\frac{z^3-z^{-3}}{2i}[/tex].

[tex]A=\frac{(z-z^{-1})(z^2-z^{-2})(z^3-z^{-3})}{-8i}[/tex]
[tex]\Rightarrow A^2=\frac{(z^2+z^{-2}-2)(z^4+z^{-4}-2)(z^6+z^{-6}-2)}{-64}[/tex]
И остава да се разкрият скобите в числителя, което ще пропусна да изпиша. Ползва се [tex]z^7=1\Rightarrow (z-1)(z^6+z^5+z^4+z^3+z^2+z+1)=0[/tex]
[tex]\Rightarrow z^6+z^5+z^4+z^3+z^2+z+1=0[/tex], понеже [tex]z\ne 1[/tex].
Получава се [tex]A^2=\frac{-7}{-64}=\frac{7}{64}[/tex], откъдето [tex]A=\frac{\sqrt{7}}{8}[/tex], понеже [tex]A>0[/tex] (трите синуса са положителни).
[tex]\Rightarrow \tan\frac{3\pi}{7}-4\sin\frac{\pi}{7}=\frac{\sin\frac{\pi}{7}\sin\frac{2\pi}{7}\sin\frac{3\pi}{7}}{\cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{3\pi}{7}}=\frac{\frac{\sqrt{7}}{8}}{\frac{1}{8}}=\sqrt{7}[/tex]