ITA 1959 ⇒ ITA 1959 - Parte II - item 5 Tópico resolvido

[petrasMOD]

Demonstrar se a afirmação é verdadeira ou falsa
[tex3]sen x+sen\ y<0[/tex3] sempre que [tex3]\frac{\pi}{2}<x<\pi[/tex3], [tex3]-\frac{\pi}{2}<y<0[/tex3] e [tex3]x-y>\pi[/tex3]
(Sugestão: Transformar soma em produto)

Resposta

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Resposta:V

[rcompany]

[tex3]

\sin x+\sin y=2\sin\left(\dfrac{x+y}{2}\right)\cos\left(\dfrac{x-y}{2}\right)\\\\

\begin{array}{rl}\left.\begin{array}{rl}
\dfrac{\pi}{2}<x<\pi\\
-\dfrac{\pi}{2}<y<0\\
x-y>\pi
\end{array}\right\}
&\implies \left\{\begin{array}{lr}
0<x+y<\pi\\
\dfrac{\pi}{2}<x-y<\dfrac{3\pi}{2}\\
x-y>\pi\end{array}\right.
\implies \left\{\begin{array}{lr}
0<x+y<\pi\\
\pi<x-y<\dfrac{3\pi}{2}\\
\end{array}\right.
\implies \left\{\begin{array}{lr}
0<\dfrac{x+y}{2}<\dfrac{\pi}{2}\\
\dfrac{\pi}{2}<x-y<\dfrac{3\pi}{4}\end{array}\right.\\
&\implies \left\{\begin{array}{lr}
0<\sin\left(\dfrac{x+y}{2}\right)<1\\
-\dfrac{\sqrt{2}}{2}<\cos\left(\dfrac{x-y}{2}\right)<0\end{array}\right.
\implies \sin x+\cos y<0\\
\end{array}

[/tex3]