Question

tg25 градусов + tg35 градусов = ?

Answer 1

$\tan\alpha+\tan\beta=\tan(\alpha+\beta)*(1-\tan\alpha\tan\beta)$

$\tan25^0+\tan35^0=\tan(25^0+35^0)*(1-\tan25^0\tan35^0)$

$\tan25^0+\tan35^0=\tan60^0*(1-\tan25^0\tan35^0)$

$\tan25^0+\tan35^0=\sqrt{3}*(1-\tan25^0\tan35^0)$

 

Отдельно вычислим произведение в скобках по формуле тангенса

$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}$

 

$\tan25^0\tan35^0=\frac{\sin25^0\sin35^0}{\cos25^0\cos35^0}$

Воспользуемся формулами произведения синусов и косинусов

$\sin\alpha\sin\beta=\frac{1}{2}*(\cos(\alpha-\beta)-cos(\alpha+\beta))$

 

$\cos\alpha\cos\beta=\frac{1}{2}*(\cos(\alpha-\beta)+cos(\alpha+\beta))$

 

$\tan25^0\tan35^0=\frac{\cos(25^0-35^0)-\cos(25^0+35^0)}{\cos(25^0-35^0)+\cos(25^0+35^0)}$

 

$\tan25^0\tan35^0=\frac{\cos10^0-\cos60^0}{\cos10^0+\cos60^0}$

 

$\tan25^0\tan35^0=\frac{\cos10^0-0,5}{\cos10^0+0,5}$

 

$1-\tan25^0\tan35^0=1-\frac{\cos10^0-0,5}{\cos10^0+0,5}$

 

$1-\frac{\cos10^0-0,5}{\cos10^0+0,5}=\frac{\cos10^0+0,5-\cos10^0+0,5}{\cos10^0+0,5}$

 

$\frac{\cos10^0+0,5-\cos10^0+0,5}{\cos10^0+0,5}=\frac{1}{\cos10^0+0,5}$

 

$\tan25^0+\tan35^0=\sqrt{3}*\frac{1}{\cos10^0+0,5}$

 

$\tan25^0+\tan35^0=\frac{\sqrt{3}}{\cos10^0+0,5}$