Question

Simplifique a expressão: y = cos 50 - cos 10 / cos 100 + cos 40

resposta: y = - √3/3

Answer 1

Identidades e propriedades que você usará neste problema:

$\star\qquad \boxed{\sf cos(A)+cos(B) =2cos\left(\dfrac{A+B}{2}\right)\cdot cos\left(\dfrac{A-B}{2}\right)}\qquad \star\\\\\\\star\qquad \boxed{\sf cos(A)-cos(B) =-2sen\left(\dfrac{A-B}{2}\right)\cdot sen\left(\dfrac{A+B}{2}\right)}\qquad\star\\\\\\\star\qquad\boxed{\sf sen(x)=cos(90^o -x)}\qquad \star$

Reorganizando e simplificando:

$\sf y = \dfrac{cos (50 ^o)- cos (10^o) }{ cos (100^o) + cos( 40^o)}\\\\\\ \sf y = \dfrac{-2sen\left(\dfrac{50^o +10^o}{2}\right)\cdot sen\left(\dfrac{50^o-10^o}{2}\right) }{2cos\left(\dfrac{100^o+40^o}{2}\right)\cdot cos\left(\dfrac{100^o-40^o}{2}\right)}\\\\\\ \sf y = \dfrac{-2sen\left(\dfrac{60^o}{2}\right)\cdot sen\left(\dfrac{40^o}{2}\right) }{2cos\left(\dfrac{140^o}{2}\right)\cdot cos\left(\dfrac{60^o}{2}\right)}\\\\\\ \sf y=\dfrac{-2sen(30^o)\cdot sen(20^o)}{2cos(70^o)\cdot cos(30^o)} \\ \\ \\ \sf y =\dfrac{-2 \cdot \dfrac{1}{2} \cdot sen(20^o)}{2cos(70^o)\cdot \dfrac{ \sqrt{3} }{2} }\\\\\\ \sf y =\dfrac{- sen(20^o)}{\sqrt{3}cos(70^o)}$

Racionalizando nossa expressão e obtemos como resultado:

$\sf y =\dfrac{- sen(20^o)}{\sqrt{3}cos(70^o)} \cdot \dfrac{\sqrt{3}}{\sqrt{3}}\\\\\\ \sf y=\dfrac{- \sqrt{3}sen(20^o)}{\sqrt{3}^2cos(70^o)} \\\\\\ \sf y= \dfrac{-\sqrt{3}sen(20^o)}{3cos(70^o)} \\\\\\ \sf y=\dfrac{-\sqrt{3}cos(90^o-20^o)}{3cos(70^o)}\\\\\\ \sf y=\dfrac{-\sqrt{3}\cancel{cos(70^o)}}{3\cancel{cos(70^o)}} \\\\\\ \boxed{\boxed{\sf y=-\dfrac{\sqrt{3}}{3}}}\quad \checkmark$