Question

Sen x = 1/2. Calcule tg x + sec x.
Com cálculo por favor.

Answer 1

Vamos partir de algo que sabemos:

$\mathtt{ sin^2(x) + cos^2(x) = 1}$

Substituindo:

$\mathtt{sin^2(x ) = [sin(x)]^2 } \\ \\ \mathtt{[sin(x)]^2 + cos^2(x) = 1} \\ \\ \mathtt{ (\frac{1}{2})^2 + cos^2(x) = 1} \\ \\ \mathtt{ cos^2 (x) = 1 - \frac{1}{4} } \\ \\ \mathtt{cos^2(x) = \frac{3}{4} } \\ \\ \mathtt{cos(x) = \frac{ \sqrt{3} }{2} }$

Logo podemos calcular:

$\mathtt{tan(x) = \frac{sin(x)}{cos(x)} \to tan(x) = \frac{1}{2} * \frac{2}{ \sqrt{3} } \to tan(x) = \frac{1}{ \sqrt{3} } \to tan(x) = \frac{ \sqrt{3} }{3} }$

$\mathtt{sec(x) = \frac{1}{cos(x)} \to sec(x) = \frac{2}{ \sqrt{3} } \to sec(x) = \frac{2 \sqrt{3} }{3} }$

Logo:

$\mathtt{tan(x) + sec(x)= \frac{ \sqrt{3} }{3} + \frac{2 \sqrt{3} }{3} = \frac{3 \sqrt{3} }{3} = \sqrt{3} }$