Question

Найдите значение выражения
$sqrt{108} cos^{2} ( frac{13 pi }{12} )- sqrt{27}$

Answer 1

$sqrt{108}cos^2frac{13pi}{12}-sqrt{27}$

$sqrt{4*27}cos^2frac{13pi}{12}-sqrt{27}$

$sqrt{27}(2cos^2frac{13pi}{12}-1})$

Далее пользуемся формулой:   $2*cos(x)-1=cos(2x)$

$sqrt{27}*cos{frac{13pi}6$

$sqrt{27}*cos{(frac{13pi}6-2pi)$

$sqrt{27}*cos{(frac{13pi}6-frac{12pi}6)$

$sqrt{27}*cos{(frac{pi}6)$

$sqrt{27}*frac{sqrt3}2$

$frac{sqrt{81}}2$

$frac{9}2$

$4,5$

Answer 2

$sqrt{108}cos^2 frac{13 pi }{12}- sqrt{27}=sqrt{108}cos^2( pi+ frac{pi}{12})}- sqrt{27}= \ sqrt{108}cos^2(-frac{pi}{12})}- sqrt{27}=sqrt{108}* frac{1+cosfrac{pi}{6}}{2}- sqrt{27}= \ sqrt{108}* frac{1+frac{ sqrt{3}}{2}}{2}- sqrt{27}=sqrt{4*27}* frac{2+ sqrt{3}}{4}}- sqrt{27}= \ sqrt{27}* frac{2+ sqrt{3}}{2}}- sqrt{27}=sqrt{27}( frac{2+ sqrt{3}}{2}}-1)=sqrt{27} frac{2+ sqrt{3}-2}{2}= frac{ sqrt{27}* sqrt{3}}{2}= \ frac{ sqrt{3*3^3}}{2}= frac{9}{2}=4.5$

Answer 3

Проверено.