Question

Найдите тангенс угла АВС

Answer 1

$BC^2 = 1 + 4^2 = 17 \\ \\ AB^2 = 1 + 4^2 = 17 \\ \\ AC^2 = 3^2 + 3^2 = 18$

Теорема косинусов
$AC^2 = BC^2 + AB^2 - 2 *BC*AB*cos(ABC) \\ \\ 18 = 17 +17-34* cos(ABC) \\ \\ 34cos(ABC) = 16 \\ \\ cos(ABC) = \frac{8}{17}$

Основное тригонометрическое тождество
cos²∠ABC + sin²∠ABC = 1
$( \frac{8}{17} )^2 + sin^2(ABC) = 1 \\ \\ sin^2(ABC) = 1- \frac{64}{289}= \frac{225}{289} \\ \\ sin(ABC) = \frac{15}{17}$

$tg(ABC) = \frac{sin(ABC)}{cos(ABC)} = \frac{15}{17} : \frac{8}{17} = \\ \\ = \frac{15}{17} * \frac{17}{8} = \frac{15}{8} =1,875$

tg∠ABC = 1,875