Тригонометрични формули

[Гост]

1. Стойността на tg(-31π върху 6) е:
2.Стойността на :
А=1+tg83°tg53° цялото върху tg83°-tg53°
3.Стойността на израза:
16tg10°cos10°cos20°cos40°

[S.B.]

3.Стойността на израза:
16tg10°cos10°cos20°cos40°

[tex]16tg10 ^\circ cos10 ^\circ cos20 ^\circ cos40 ^\circ =[/tex]

[tex]= 16 \frac{sin10 ^\circ }{cos10 ^\circ } . \frac{2.sin10 ^\circ cos10 ^\circ }{2sin10 ^\circ }.cos20 ^\circ.cos40 ^\circ =[/tex]

[tex]= \frac{8}{cos10 ^\circ }.sin20 ^\circ cos20 ^\circ cos40 ^\circ = \frac{8}{cos10 ^\circ }. \frac{2sin20 ^\circ cos20 ^\circ }{2}.cos40 ^\circ =[/tex]

[tex]= \frac{4}{cos10 ^\circ }.sin40 ^\circ.cos40 ^\circ = \frac{4}{cos10 ^\circ }. \frac{2sin40 ^\circ cos40 ^\circ }{2} =[/tex]

[tex]= \frac{2}{cos10 ^\circ }.sin80 ^\circ = \frac{2}{cos10 ^\circ }.sin(90 ^\circ - 10 ^\circ ) =[/tex]

[tex]= \frac{2}{cos10 ^\circ }.cos10 ^\circ = 2[/tex]

[S.B.]

1. Стойността на tg(-31π върху 6) е:

По дефиниция [tex]tgx = \alpha _{0 } + k \pi[/tex]

[tex]tg(- \frac{31 \pi }{6} ) = tg( - \frac{30 \pi }{6} - \frac{ \pi }{6}) = tg[- (\frac{ \pi }{6} + 5 \pi )] = tg( - \frac{ \pi }{6}) = - tg \frac{ \pi }{6} = - \frac{ \sqrt{3} }{3}[/tex]

[S.B.]

2.Стойността на :
А=1+tg83°tg53° цялото върху tg83°-tg53°

[tex]A =\displaystyle \frac{1 + tg83 ^\circ tg53 ^\circ }{tg83 ^\circ - tg53 ^\circ } = \displaystyle \frac{1 + \displaystyle\frac{sin83}{cos83} .\displaystyle \frac{sin53 ^\circ }{cos53 ^\circ } }{\displaystyle \frac{sin83 ^\circ }{cos83 ^\circ } - \displaystyle \frac{sin53 ^\circ }{cos53 ^\circ } } = \displaystyle \frac{\displaystyle \frac{cos83 ^\circ cos53 ^\circ + sin83 ^\circ sin53 ^\circ }{cos83 ^\circ cos53 ^\circ } }{\displaystyle \frac{sin83 ^\circ cos53 ^\circ - cos83 ^\circ sin53 ^\circ }{cos53 ^\circ cos83 ^\circ } } = \displaystyle \frac{cos(83 ^\circ - 53 ^\circ) }{sin(83 ^\circ - 53 ^\circ) } =[/tex]

[tex]= \frac{cos30 ^\circ }{sin30 ^\circ } = cotg30 ^\circ = \sqrt{3}[/tex]