Question

Se sen x + cos x = 1/5, com 0 < (igual) x < (igual) pi, então o valor de sen 2x é:
a) - 12/25
b) - 24/25
c) 12/25
d) 16/25
e) 24/25

Answer 1

$\mathrm{sen\,}x+\cos x=\frac{1}{5}$


Elevando os dois lados ao quadrado, temos

$(\mathrm{sen\,}x+\cos x)^{2}=\left(\frac{1}{5} \right )^{2}\\ \\ \mathrm{sen^{2}\,}x+2\,\mathrm{sen\,}x\cos x+\cos^{2} x=\frac{1}{25}\\ \\ \left(\mathrm{sen^{2}\,}x+\cos^{2} x \right )+2\,\mathrm{sen\,}x\cos x=\frac{1}{25}\\ \\ 1+2\,\mathrm{sen\,}x\cos x=\frac{1}{25}\\ \\ 2\,\mathrm{sen\,}x\cos x=\frac{1}{25}-1\\ \\ 2\,\mathrm{sen\,}x\cos x=\frac{1-25}{25}\\ \\ 2\,\mathrm{sen\,}x\cos x=-\frac{24}{25}\\ \\ \boxed{\begin{array}{c}\mathrm{sen\,}2x=-\frac{24}{25} \end{array}}$


Resposta: alternativa $\text{b) }-24/25.$