Question

se 10 sen² x + 14 cos²x = 11 , obter os possíveis valores positivos de tg x.

Answer 1

Resposta:

Explicação passo-a-passo:

$10sen^{2}x+14cos^{2}x=11$

$10sen^{2}x+10cos^{2}x+4cos^{2}x=11$

$10(sen^{2}x+cos^{2}x)+4cos^{2}x=11$

$10.1+4cos^{2}x=11$

$10+4cos^{2}x=11$

$4cos^{2}x=11-10$

$4cos^{2}x=1$

$cos^{2}x=\dfrac{1}{4}\\\\cosx=\sqrt{\dfrac{1}{4}}$

$\boxed{cosx={\dfrac{1}{2}}}$

$sen^{2}x+cos^{2}x=1\\\\sen^{2}x+(\dfrac{1}{2})^{2}=1\\\\sen^{2}x+\dfrac{1}{4}=1\\\\sen^{2}x=1-\dfrac{1}{4}\\\\sen^{2}x=\dfrac{4}{4}-\dfrac{1}{4}\\\\sen^{2}x=\dfrac{3}{4}\\\\senx=\sqrt{\dfrac{3}{4}}\\\\\boxed{senx=\dfrac{\sqrt{3}}{2}}$

$tgx=\dfrac{senx}{cosx}\\\\\\tgx=\dfrac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\\\\\\tgx=\frac{\sqrt{3}}{2}.2\\ \\\boxed{tgx=\sqrt{3}}$