Question

Se sen x – cos x = 1/2, então sen 2x é:

a) 1/4
b) -3/4
c) 3/4
d) 1/2
e) √3/4

Answer 1

Resposta:

$\text{letra C}$

Explicação passo-a-passo:

$\begin{cases}\text{sen}^2x + \text{cos}^2x = 1\\\text{sen }x - \text{cos }x = \dfrac{1}{2}\end{cases}$

$\text{sen }x = \dfrac{1}{2} + \text{cos }x$

$(\dfrac{1}{2} + \text{cos}\: x)^2 + \text{cos}^2x = 1$

$(\dfrac{1}{4} + \text{cos x} + \text{cos}^2x) + \text{cos}^2x = 1$

$2\text{ cos}^2\:x + \text{cos }x - \dfrac{3}{4} = 0$

$\Delta = b^2 - 4ac$

$\Delta = 1^2 - 4.2.(-\dfrac{3}{4})$

$\Delta = 1 + 6$

$\Delta = 7$

$x = \dfrac{-b \pm \sqrt{\Delta}}{2a}$

$x = \dfrac{-1 + \sqrt{7}}{2.2} = \dfrac{-1 + \sqrt{7}}{4}$

$\boxed{\boxed{\text{cos }x = \dfrac{-1 + \sqrt{7}}{4}}}$

$\text{sen }x = \dfrac{1}{2} + \dfrac{-1 + \sqrt{7}}{4}$

$\text{sen }x = \dfrac{2 -1 + \sqrt{7}}{4}$

$\boxed{\boxed{\text{sen }x = \dfrac{1 + \sqrt{7}}{4}}}$

$\text{sen } 2x = 2 \times \text{sen }x \times \text{ cos }x$

$\text{sen } 2x = 2 \times \dfrac{1 + \sqrt{7}}{4} \times \dfrac{-1 + \sqrt{7}}{4}$

$\text{sen } 2x = \dfrac{-1 + \sqrt{7} - + \sqrt{7} + 7}{8} = \dfrac{6}{8}$

$\boxed{\boxed{\text{sen } 2x = \dfrac{3}{4}}}$

Answer 2

Resposta:

sen(2x) = 3/4

Explicação passo-a-passo:

vamos primeiramente lembrar a fórmula do sen(2x)

sen(2x) = 2sen(x)cos(x)

$\left \{ {{sen(x)^{2}+cos(x)^{2} =1} \atop {sen(x) - cos(x)=\frac{1}{2} }} \right.$

elevando a 2 equação ao quadrado

$sen(x)^{2} + cos(x)^{2} -2sen(x)cos(x) = \frac{1}{4} \\\\1 - sen(2x) = \frac{1}{4} \\sen(2x) = \frac{3}{4}$