Question

Se tgx = √7 e x ∈ [π, 3π 2 ], então sen3x é igual a



A ( ) −√14/8
. B ( )√14/8
. C ( )√14/4
. D ( ) −√14/4
. E ( )√14/6
.

Answer 1

Olá Eversonboy.



Identidades trigonométricas usadas:


$\star~~\boxed{\boxed{\mathsf{sen^2x+cos^2x=1}}}\\\\\\\star~~\boxed{\boxed{\mathsf{tg~x=\dfrac{sen~x}{cos~x}}}}$

$\star~~\boxed{\boxed{\mathsf{sen(3x)=3sen~x-4sen^3x}}}$


Organizando as informações e resolvendo a equação.

$\mathsf{tg~x=\sqrt{7}\qquad\qquad\qquad\qquad x\in\Big[\pi,\dfrac{3\pi}{2}\mathsf{\Big]}}\\\\\\\\\\\mathsf{\sqrt{7}=\dfrac{sen~x}{cos~x}}\\\\\\\mathsf{\sqrt{7}=\dfrac{sen~x}{\sqrt{1-sen^2x}}}\\\\\\\mathsf{\sqrt{7-7sen^2x}=sen~x}\\\\\mathsf{7-7sen^2x=sen^2x}\\\\\mathsf{7-8sen^2x=0}\\\\\mathsf{-8sen^2x=-7~\cdot(-1)}\\\\\mathsf{sen^2x=\dfrac{7}{8}}\\\\\mathsf{sen~x=\pm\sqrt{\dfrac{7}{8}}}$

Chegamos a 2 soluções. Precisamos verificar o intervalo para saber qual é a válida.

$\mathsf{\pi \leq x \leq \dfrac{3\pi}{2}}\\\\\\\mathsf{sen~\pi\geq sen~x\geq sen~\dfrac{3\pi}{2}}\\\\\\\mathsf{0\geq sen~x\geq -1}\\\\\\\mathsf{sen~x=-\sqrt{\dfrac{7}{8}}~\checkmark}$

$\mathsf{sen(3x)=3\mathsf{sen~x} - 4sen^3x }\\\\\mathsf{sen(3x)=3\cdot\Big(-\sqrt{\dfrac{7}{8}}\Big)-4\cdot\Big(-\sqrt{\dfrac{7}{8}}\Big)^3}\\\\\\\mathsf{sen(3x)=-\dfrac{3\sqrt{7}}{\sqrt{8}}+4\cdot\Big(\dfrac{7\sqrt{7}}{8\sqrt{8}}\Big)}\\\\\\\mathsf{sen(3x)=-\dfrac{3\sqrt{7}}{\sqrt{8}}+\dfrac{7\sqrt{7}}{2\sqrt{8}}}\\\\\\\mathsf{sen(3x)=-\dfrac{6\sqrt{7}}{2\sqrt{8}}+\dfrac{7\sqrt{7}}{2\sqrt{8}}}\\\\\\\mathsf{sen(3x)=\dfrac{\sqrt{7}}{2\sqrt{8}}\cdot\dfrac{\sqrt{8}}{\sqrt{8}}}$

$\mathsf{sen(3x)=\dfrac{\sqrt{56}}{2\cdot8}}\\\\\\\mathsf{sen(3x)=\dfrac{\sqrt{2^2\cdot 2\cdot7}}{16}}\\\\\\\mathsf{sen(3x)=\dfrac{\diagup\!\!\!\!2\sqrt{14}}{\diagup\!\!\!\!16}}\\\\\\\boxed{\mathsf{sen(3x)=\dfrac{\sqrt{14}}{8}}}$


Resposta (b)



Dúvidas? comente.