Question

Calcule o valor exato de:
sen(3π/8)^2-cos(3π/8)^2

Answer 1

Para esta tarefa, vamos utilizar uma das identidades para o cosseno do arco duplo:

    $\mathsf{cos^2(\theta)-sen^2(\theta)=cos(2\theta)}$

para todo $\theta\in\mathbb{R}.$

Reescrevendo a expressão e aplicando a identidade acima para $\mathsf{\theta=\dfrac{3\pi}{8},}$  obtemos

    $\mathsf{sen^2\!\left(\dfrac{3\pi}{8}\right)-cos^2\!\left(\dfrac{3\pi}{8}\right)}\\\\\\ \mathsf{=-\left[-\,sen^2\!\left(\dfrac{3\pi}{8}\right)+cos^2\!\left(\dfrac{3\pi}{8}\right)\right]}\\\\\\ \mathsf{=-\left[cos^2\!\left(\dfrac{3\pi}{8}\right)-sen^2\!\left(\dfrac{3\pi}{8}\right)\right]}\\\\\\ \mathsf{=-\,cos\!\left(2\cdot \dfrac{3\pi}{8}\right)}\\\\\\ \mathsf{=-\,cos\!\left(\diagup\!\!\!\! 2\cdot \dfrac{3\pi}{\diagup\!\!\!\! 2\cdot 4}\right)}\\\\\\ \mathsf{=-\,cos\!\left(\dfrac{3\pi}{4}\right)}$

    $\mathsf{=-\bigg(\!\!\!-\dfrac{\sqrt{2}}{2}\bigg)}\\\\\\ \mathsf{=\dfrac{\sqrt{2}}{2}\quad\longleftarrow\quad resposta.}$

Dúvidas? Comente.

Bons estudos! :-)