Question

(9 класс)
2. Упростить:
(tg3˚ tg57˚ tg63˚)(tg17˚ tg43˚ tg73˚)(tg23˚ tg37˚ tg83˚)

Answer 1

$\Big(tg3\textdegree~tg57\textdegree~tg63\textdegree \Big)\Big(tg17\textdegree~tg43\textdegree~tg73\textdegree \Big)\Big(tg23\textdegree~tg37\textdegree~tg83\textdegree \Big)=\\=\Big(tg17\textdegree~tg43\textdegree~tg\big(90\textdegree-17\textdegree\big)\Big) \Big(tg3\textdegree~tg\big(60\textdegree-3\textdegree\big) ~tg\big(60\textdegree+3\textdegree\big)\Big)\times$

$~~~~~~~~~~~~~~~~\times \Big(tg23\textdegree~tg\big(60\textdegree-23\textdegree\big) ~tg\big(60\textdegree+23\textdegree\big)\Big)=\\=\Big(\underbrace{tg17\textdegree~ctg17\textdegree}_{=1}~tg43\textdegree\Big) \Big(tg3\textdegree~\cdot \dfrac{tg60\textdegree-tg3\textdegree}{1+tg60\textdegree\cdot tg3\textdegree}\cdot \dfrac{tg60\textdegree+tg3\textdegree}{1-tg60\textdegree\cdot tg3\textdegree}\Big)\times$

$~~~~~~~~~~~~~~~~\times \bigg(tg23\textdegree\cdot \dfrac{tg60\textdegree-tg23\textdegree}{1+tg60\textdegree\cdot tg23\textdegree}\cdot \dfrac{tg60\textdegree+tg23\textdegree}{1-tg60\textdegree\cdot tg23\textdegree}\bigg)=$

$=tg43\textdegree\cdot \bigg(tg3\textdegree~\cdot \dfrac{tg^260\textdegree-tg^23\textdegree}{1-tg^260\textdegree\cdot tg^23\textdegree}\bigg)\bigg(tg23\textdegree\cdot \dfrac{tg^260\textdegree-tg^223\textdegree}{1-tg^260\textdegree\cdot tg^223\textdegree}\bigg)=$

$=tg43\textdegree\cdot \bigg(tg3\textdegree~\cdot \dfrac{\big(\sqrt3\big)^2-tg^23\textdegree}{1-\big(\sqrt3\big)^2\cdot tg^23\textdegree}\bigg)\bigg(tg23\textdegree\cdot \dfrac{\big(\sqrt3\big)^2-tg^223\textdegree}{1-\big(\sqrt3\big)^2\cdot tg^223\textdegree}\bigg)=$

$=tg43\textdegree\cdot \bigg(tg3\textdegree~\cdot \dfrac{3-tg^23\textdegree}{1-3tg^23\textdegree}\bigg)\bigg(tg23\textdegree\cdot \dfrac{3-tg^223\textdegree}{1-3tg^223\textdegree}\bigg)=\\\\=tg43\textdegree\cdot \bigg(\dfrac{3tg3\textdegree-tg^33\textdegree}{1-3tg^23\textdegree}\bigg)\bigg(\dfrac{3tg23\textdegree-tg^323\textdegree}{1-3tg^223\textdegree}\bigg)=$

$=tg43\textdegree\cdot \bigg(tg\big(3\cdot3\textdegree\big)\bigg)\bigg(tg\big(3\cdot23\textdegree\big)\bigg)\boldsymbol{=tg43\textdegree\cdot tg9\textdegree\cdot tg69\textdegree}$

Здесь можно закончить. Если продолжить дальше :

$tg43\textdegree\cdot tg9\textdegree\cdot tg\big(60\textdegree+9\textdegree\big)=tg43\textdegree\cdot tg9\textdegree\cdot \dfrac {tg60\textdegree+tg9\textdegree}{1-tg60\textdegree\cdot tg9\textdegree}=$

$=tg43\textdegree\cdot tg9\textdegree\cdot \dfrac {\sqrt3+tg9\textdegree}{1-\sqrt3\cdot tg9\textdegree}=\\=tg43\textdegree\cdot tg9\textdegree\cdot \dfrac {\big(\sqrt3+tg9\textdegree\big)\big(\sqrt3-tg9\textdegree\big)\big(1+\sqrt3\cdot tg9\textdegree\big)}{\big(1-\sqrt3\cdot tg9\textdegree\big)\big(1+\sqrt3\cdot tg9\textdegree\big)\big(\sqrt3-tg9\textdegree\big)}=$

$=tg43\textdegree\cdot tg9\textdegree\cdot \dfrac {\big(3-tg^29\textdegree\big)\big(1+\sqrt3\cdot tg9\textdegree\big)}{\big(1-3tg^29\textdegree\big)\big(\sqrt3-tg9\textdegree\big)}=\\=tg43\textdegree\cdot \dfrac {3tg9\textdegree-tg^39\textdegree}{1-3tg^29\textdegree}\cdot \dfrac {1+\sqrt3\cdot tg9\textdegree}{\sqrt3-tg9\textdegree}=$$=tg43\textdegree\cdot tg\big(3\cdot 9\textdegree\big)\cdot \dfrac {1+\sqrt3\cdot tg9\textdegree}{\sqrt3-tg9\textdegree}=tg43\textdegree\cdot tg27\textdegree\cdot \dfrac {1+\sqrt3\cdot tg9\textdegree}{\sqrt3-tg9\textdegree}$

============================================

Использованы формулы

$tg\big(\alpha +\beta \big)=\dfrac {tg\alpha +tg\beta }{1-tg\alpha \cdot tg\beta}\\\\tg\big(\alpha -\beta \big)=\dfrac {tg\alpha -tg\beta }{1+tg\alpha \cdot tg\beta}\\\\tg\big(3\alpha \big)=\dfrac{3tg\alpha -tg^3\alpha }{1-3tg^2\alpha }$