Question

Если:
$\cos(a) ) = \frac{4 \sqrt{3} }{7}$
и 0<а<90°

найдите tga/2=?
​

Answer 1

Ответ:   $tg\dfrac{a}{2}=7-4\sqrt3$  .

$cosa=\dfrac{4\sqrt3}{7}\ \ ,\qquad \boxed{sin^2a+cos^2a=1\ }\\\\\\0^\circ < a < 90^\circ \ \ \ \Rightarrow \ \ \ sina > 0\ \ ,\ \ sina=+\sqrt{1-cos^2a}\\\\\\sina=\sqrt{1-\dfrac{16\cdot 3}{49}}=\sqrt{\dfrac{1}{49}}=\dfrac{1}{7}\\\\\\tg\dfrac{a}{2}=\dfrac{sina}{1+cosa}=\dfrac{\dfrac{1}{7}}{1+\dfrac{4\sqrt3}{7}}=\dfrac{1}{7+4\sqrt3}=\dfrac{7-4\sqrt3}{(7+4\sqrt3)(7-4\sqrt3)}=\dfrac{7-4\sqrt3}{49-48}=\\\\=7-4\sqrt3$