Тригонометричен израз

[emil3993]

Здравейте математици! Може ли решението? Благодаря предварително. Приятен ден.

[KOPMOPAH]

$\frac{2tg15^{\circ}}{1+tg^215^{\circ}}+\frac{2tg15^{\circ}}{1-tg^215^{\circ^2}}=\frac{2tg15^{\circ}}{1+tg^215^{\circ}}.\frac{cos^215^{\circ}}{cos^215^{\circ}}+\frac{2tg15^{\circ}}{1-tg^215^{\circ^2}}.\frac{cos^215^{\circ}}{cos^215^{\circ}}$ ... нататък сещаш ли се

[Nathi123]

[tex]\frac{2tg15^\circ}{1+tg^{2}15^\circ} + \frac{2tg15^\circ}{1 - tg^{2}15^\circ}[/tex] =[tex]sin30^\circ + tg30^\circ =\frac{1}{2} + \frac{\sqrt{3}}{3}=\frac{3+2\sqrt{3}}{6}[/tex] .
В решението се използват формулите [tex]sin2\alpha = \frac{2tg\alpha}{1+tg^{2}\alpha} ; tg2\alpha = \frac{2tg\alpha}{1-tg^{2}\alpha} .[/tex]

[mail_dinko]

[tex]tg 15 = \sqrt {\frac {1-cos 30}{1+ cos 30}}[/tex]
[tex]\frac{2 tg15}{1+ tg^2 15}+\frac{2 tg 15}{1- tg^2 15}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1+ ( \sqrt {\frac {1-cos 30}{1+ cos 30}})^2}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1- ( \sqrt {\frac {1-cos 30}{1+ cos 30}})^2}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1+ \frac {1-cos 30}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1- \frac {1-cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1+cos 30}{1+ cos 30}+ \frac {1-cos 30}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1+cos 30}{1+ cos 30}- \frac {1-cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1 \cancel {+cos 30}+1 \cancel {-cos 30}}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {\cancel {1+}cos 30 \cancel {-1}+ cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac {\cancel {2 .} \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{\cancel {2}} + \frac {\cancel {2 .} \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{\cancel {2} cos 30}=[/tex]
[tex]=\sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30) + \frac { \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{ cos 30}=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2}) + \frac { \sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+ \frac {\sqrt {3}}{2}}} (1 + \frac {\sqrt {3}}{2})}{ \frac {\sqrt {3}}{2}}=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2})(1+ \frac {2}{\sqrt {3}})=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2})(1+ \frac {2}{\sqrt {3}})=[/tex]
[tex]= \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {1+\frac {\sqrt {3}}{2}}}(1 + \frac {\sqrt {3}}{2}+ \frac {2}{\sqrt {3}}+1)=[/tex]
[tex]= \frac {\cancel {\frac {1}{2} } (\sqrt {3}-1)}{\cancel { \frac {1}{2}} (\sqrt {3}+1)}(2 + \frac {3+4}{2\sqrt {3}})= \frac {\sqrt {3}-1}{\sqrt {3}+1 } . \frac {\sqrt {3}-1}{\sqrt {3}-1}(2 + \frac {7}{2\sqrt {3}} . \frac {\sqrt {3}}{\sqrt {3}})=[/tex]
[tex]=\frac {3+1- 2 \sqrt {3}}{2} (2 + \frac {7 \sqrt {3}}{6})= \frac {4-2 \sqrt {3}}{2}. \frac {12 + 7 \sqrt {3}}{6}=[/tex]
[tex]=\frac {(2- \sqrt {3})(12 + 7 \sqrt {3})}{6}=\frac {24+14 \sqrt {3} - 12 \sqrt {3} - 21}{6}=[/tex]
[tex]= \frac {3 + 2 \sqrt {3}}{6}[/tex]

[emil3993]

[tex]tg 15 = \sqrt {\frac {1-cos 30}{1+ cos 30}}[/tex]
[tex]\frac{2 tg15}{1+ tg^2 15}+\frac{2 tg 15}{1- tg^2 15}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1+ ( \sqrt {\frac {1-cos 30}{1+ cos 30}})^2}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1- ( \sqrt {\frac {1-cos 30}{1+ cos 30}})^2}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1+ \frac {1-cos 30}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{1- \frac {1-cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1+cos 30}{1+ cos 30}+ \frac {1-cos 30}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1+cos 30}{1+ cos 30}- \frac {1-cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {1 \cancel {+cos 30}+1 \cancel {-cos 30}}{1+ cos 30}}+\frac{2 . \sqrt {\frac {1-cos 30}{1+ cos 30}}}{\frac {\cancel {1+}cos 30 \cancel {-1}+ cos 30}{1+ cos 30}}=[/tex]
[tex]=\frac {\cancel {2 .} \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{\cancel {2}} + \frac {\cancel {2 .} \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{\cancel {2} cos 30}=[/tex]
[tex]=\sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30) + \frac { \sqrt {\frac {1-cos 30}{1+ cos 30}} (1 + cos 30)}{ cos 30}=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2}) + \frac { \sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+ \frac {\sqrt {3}}{2}}} (1 + \frac {\sqrt {3}}{2})}{ \frac {\sqrt {3}}{2}}=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2})(1+ \frac {2}{\sqrt {3}})=[/tex]
[tex]=\sqrt {\frac {1-\frac {\sqrt {3}}{2}}{1+\frac {\sqrt {3}}{2}}} (1 +\frac {\sqrt {3}}{2})(1+ \frac {2}{\sqrt {3}})=[/tex]
[tex]= \frac {\sqrt {1-\frac {\sqrt {3}}{2}}}{\sqrt {1+\frac {\sqrt {3}}{2}}}(1 + \frac {\sqrt {3}}{2}+ \frac {2}{\sqrt {3}}+1)=[/tex]
[tex]= \frac {\cancel {\frac {1}{2} } (\sqrt {3}-1)}{\cancel { \frac {1}{2}} (\sqrt {3}+1)}(2 + \frac {3+4}{2\sqrt {3}})= \frac {\sqrt {3}-1}{\sqrt {3}+1 } . \frac {\sqrt {3}-1}{\sqrt {3}-1}(2 + \frac {7}{2\sqrt {3}} . \frac {\sqrt {3}}{\sqrt {3}})=[/tex]
[tex]=\frac {3+1- 2 \sqrt {3}}{2} (2 + \frac {7 \sqrt {3}}{6})= \frac {4-2 \sqrt {3}}{2}. \frac {12 + 7 \sqrt {3}}{6}=[/tex]
[tex]=\frac {(2- \sqrt {3})(12 + 7 \sqrt {3})}{6}=\frac {24+14 \sqrt {3} - 12 \sqrt {3} - 21}{6}=[/tex]
[tex]= \frac {3 + 2 \sqrt {3}}{6}[/tex]

От любопитство сам ли го ниписа или го генерира с програма?
Благодаря за решенията.

[mail_dinko]

Във форума има възможност за изписване на формули и дроби и т.н. - виж бутона TEX.

[emil3993]

Благодаря.