Question

10) Sendo x 30 calcule o valor da expressão:
y= 2 senx - 4 cosx + tg 2x / cos 2x - sen 2x

Answer 1

numerador:
2 sen x-4 cos x+tg 2x
2. sen30 - 4.cos30 + tg(2.30)
2.1/2 - 4.√3/2 + tg 60
1 - 2√3 + √3
1-√3 >

denominador:
cos 4x-sen 2x
cos 4.30 - sen 2.30
cos 120 - sen 60
-1/2 - √3/2
-(1+√3)/2 >

y=2 sen x-4 cos x+tg 2x/cos 4x-sen 2x
y = (1-√3) / -(1+√3)/2
y = (1-√3) . -2/(1+√3)
y = -2. (1-√3) / (1+√3).... *(1-√3)/(1-√3)
y= -2.(1-√3)² / (1-3)
y= -2.(1-√3)² / -2
y = (1-√3)²
y= 1+3 -2√3
y= 4-2√3
y= 2(2-√3)

Answer 2

Cálculo de y, considerando que x = 30º:

$y = \frac{2senx - 4cosx+tg2x}{cos2x-sen2x}$

Lembrando que:
$sen30 = \frac{1}{2}\\cos30 = \frac{ \sqrt{3} }{2}\\tg30= \frac{ \sqrt{3} }{3}$

Vamos por partes:
$2sen30 = 2* \frac{1}{2} = 1$
$4cos30 = 4*\frac{ \sqrt{3} }{2} = 2 \sqrt{3}$
$tg2x = tg2*30 = tg60 = \sqrt[]{3}$
$cos2x = cos2*30 = cos60 = \frac{1}{2}$
$sen2x = sen2*30 = sen60 = \frac{ \sqrt[]{3} }{2}$

Aplicando na fórmula:
$y = \frac{1 - 2 \sqrt[]{3} + \sqrt[]{3}}{\frac{1}{2} - \frac{ \sqrt[]{3} }{2}}$
$y = \frac{1 - \sqrt[]{3}}{\frac{ 1- \sqrt[]{3} }{2}}$
$y = \frac{2*(1 - \sqrt[]{3})}{1- \sqrt[]{3}}$
$y = 2$