Пресметнете без помощта на таблица

[Гост]

кос(2п/7) + кос(4п/7) + кос(6п/7) + 1/2
помогнете , че не мога да го измъдря..

[S.B.]

[tex]\cos \frac{2 \pi }{7} + \cos \frac{4 \pi }{7} + \cos \frac{6 \pi }{7} + \frac{1}{2} =[/tex]

[tex]= \cos \frac{2 \pi }{7} + \cos( \frac{7 \pi }{7}- \frac{3 \pi }{7}) + \cos( \frac{7 \pi }{7} - \frac{ \pi }{7}) + \frac{1}{2} =[/tex]

[tex]= \cos \frac{2 \pi }{7} - \cos \frac{3 \pi }{7} - \cos \frac{ \pi }{7} + \frac{1}{2} =[/tex]

[tex]= - ( \cos \frac{ \pi }{7} - \cos \frac{2 \pi }{7} + \cos \frac{3 \pi }{7}) + \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{2\sin \displaystyle \frac{ \pi }{7}(\cos \displaystyle \frac{ \pi }{7} - \cos \displaystyle \frac{2 \pi }{7} + \cos \displaystyle \frac{3 \pi }{7} )}{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{2\sin \displaystyle \frac{ \pi }{7}\cos \displaystyle \frac{ \pi }{7} - 2\sin \displaystyle \frac{ \pi }{7}\cos \displaystyle \frac{2 \pi }{7} + 2\sin\displaystyle \frac{ \pi }{7}\cos \displaystyle \frac{3 \pi }{7} }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{\sin \displaystyle \frac{2 \pi }{7} - 2.\displaystyle \frac{1}{2}(\sin \displaystyle \frac{ \pi + 2 \pi }{7} + \sin\displaystyle \frac{ \pi - 2 \pi }{7}) + 2.\displaystyle \frac{1}{2}(\sin \displaystyle \frac{ \pi + 3 \pi }{7} + \sin \displaystyle \frac{ \pi - 3 \pi }{7}) }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{\sin \displaystyle \frac{2 \pi }{7} - \sin \displaystyle \frac{3 \pi }{7} + \sin \displaystyle \frac{ \pi }{7} + \sin \displaystyle \frac{4 \pi }{7} - \sin \displaystyle \frac{2 \pi }{7} }{2\sin\displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{ - \sin \displaystyle \frac{3 \pi }{7} + \sin \displaystyle \frac{ \pi }{7} + \sin \displaystyle \frac{4 \pi }{7} }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{-\sin\displaystyle \frac{3 \pi }{7} + \sin \displaystyle \frac{ \pi }{7} + \sin (\displaystyle \frac{7 \pi }{7} - \displaystyle \frac{3 \pi }{7}) }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{-\sin \displaystyle \frac{3 \pi }{7} + \sin \displaystyle \frac{ \pi }{7} + \sin \displaystyle \frac{3 \pi }{7} }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \displaystyle \frac{\sin \displaystyle \frac{ \pi }{7} }{2\sin \displaystyle \frac{ \pi }{7} } + \displaystyle \frac{1}{2} =[/tex]

[tex]= - \frac{1}{2} + \frac{1}{2} = 0[/tex]

[Гост]

благодаря , ако може и
тг(9*) - тг(27*) - тг(63*) + тг(81*)

[Евва]

Ще използвам формулата tg[tex]\alpha[/tex]-tg[tex]\beta[/tex]=[tex]\frac{sin( \alpha- \beta) }{cos \alpha.cos \beta }[/tex]

(tg9[tex]^\circ[/tex]-tg63[tex]^\circ[/tex])+(tg81[tex]^\circ[/tex]-tg27[tex]^\circ[/tex])=

=[tex]\frac{sin(9 ^\circ -63 ^\circ) }{cos9 ^\circ.cos63 ^\circ }[/tex]+[tex]\frac{sin(81 ^\circ-27 ^\circ )}{cos81 ^\circ .cos27 ^\circ }[/tex]=

=[tex]\frac{sin(-54 ^\circ )}{cos9 ^\circ.cos63 ^\circ }[/tex]+[tex]\frac{sin54 ^\circ }{cos(90 ^\circ-9 ^\circ).cos(90 ^\circ-63 ^\circ) }[/tex]=

=[tex]\frac{- sin54 ^\circ }{cos9 ^\circ.cos63 ^\circ }[/tex]+[tex]\frac{sin54 ^\circ }{sin9 ^\circ .sin63 ^\circ }[/tex]=

=[tex]\frac{2.2}{2.2}[/tex] .[tex]\frac{sin54 ^\circ .cos9 ^\circ .cos63 ^\circ -sin54 ^\circ.sin9 ^\circ.sin63 ^\circ }{sin9 ^\circ.cos9 ^\circ .sin63 ^\circ.cos63 ^\circ }[/tex]=

=[tex]\frac{4sin54 ^\circ[ cos(9 ^\circ+63 ^\circ) ] }{sin18 ^\circ.sin126 ^\circ }[/tex] използваме ,че 126[tex]^\circ[/tex]=90[tex]^\circ[/tex]+36[tex]^\circ[/tex]

=[tex]\frac{4sin54 ^\circ .cos72 ^\circ }{sin18 ^\circ.cos36 ^\circ }[/tex]=

[tex]\frac{4sin54 ^\circ.cos(90 ^\circ -18 ^\circ )}{sin18 ^\circ.cos(90 ^\circ -54 ^\circ )}[/tex]=

=[tex]\frac{4sin54 ^\circ.sin18 ^\circ }{sin18 ^\circ.sin54 ^\circ }[/tex]=

= 4