Question

Sendo x tal que 0 ≤ x ≤ π/2 e sen x = 7/17. Determine o valor da sec x

Answer 1

$\mathrm{Seja}\ x\in{\left[0,\dfrac{\pi}{2}\right]}\ \mathrm{tal\ que}\ \sin{x}=\dfrac{7}{17}.$

$\mathrm{O}\ \cos{x}\ \mathrm{pode\ ser\ obtido\ pela\ seguinte\ rela\c{c}\tilde{a}o}\text{:}$

$\sin^2{x}+\cos^2{x}=1\Longrightarrow \cos{x}=\pm\sqrt{1-\sin^2{x}}$

$\Longrightarrow\cos{x}=\pm\sqrt{1-\bigg(\dfrac{7}{17}\bigg)^2}=\pm\sqrt{\dfrac{240}{289}}\Longrightarrow \cos{x}=\pm\dfrac{4\sqrt{15}}{17}$

$\Longrightarrow \cos{x}=\dfrac{4\sqrt{15}}{17}\ \because\ 0\leq x\leq\dfrac{\pi}{2}$

$\mathrm{Desse\ modo,\ a}\ \sec{x}\ \mathrm{ser\acute{a}}\text{:}$

$\sec{x}=\dfrac{1}{\cos{x}}\Longrightarrow\sec{x}=\dfrac{1}{\frac{4\sqrt{15}}{17}}=\dfrac{17}{4\sqrt{15}}\ \therefore\ \boxed{\sec{x}=\dfrac{17\sqrt{15}}{60}}$