Question

Sabendo que senx + cosx = 1/2 , determine sen (2x)

Answer 1

$\mathrm{sen\,}{x}+\cos{x}=\frac{1}{2}\\ \\ \left(\mathrm{sen\,}{x}+\cos{x} \right )^{2}=\frac{1}{4}\\ \\ \mathrm{sen^{2}\,}{x}+2\,\mathrm{sen\,}{x}\cdot \cos{x}+\cos^{2}{x}=\frac{1}{4}\\ \\ \underbrace{2\,\mathrm{sen\,}{x}\cdot \cos{x}}_{\mathrm{sen\,}{\left(2x \right )}}+\underbrace{\mathrm{sen^{2}\,}{x}+\cos^{2}{x}}_{1}=\frac{1}{4}\\ \\ \mathrm{sen\,}{\left(2x \right )}+1=\frac{1}{4}\\ \\ \mathrm{sen\,}{\left(2x \right )}=\frac{1}{4}-1\\ \\ \mathrm{sen\,}{\left(2x \right )}=\frac{1-4}{4}\\ \\ \boxed{\mathrm{sen\,}{\left(2x \right)=-\frac{3}{4}}}$