Question

Найти cos α, tg x и ctg x,если $sin \alpha = \frac{7}{25}$ и $\alpha$ ∈$(0;\frac{\pi }{2})$ .

Answer 1

$\alpha\in\Big(0 \ ; \ \dfrac{\pi }{2}\Big) \ \Rightarrow \ Cos\alpha 0 \ , \ tg\alpha 0 \ , \ Ctg\alpha0 \\\\Sin\alpha=\dfrac{7}{25}\\\\Cos\alpha=\sqrt{1-Sin^{2}\alpha} =\sqrt{1-\Big(\dfrac{7}{25}\Big)^{2}}=\sqrt{1-\dfrac{49}{625} }=\sqrt{\dfrac{576}{625} } =\dfrac{24}{25} \\\\\\tg\alpha=\dfrac{Sin\alpha }{Cos\alpha } =\dfrac{7}{25}:\dfrac{24}{25}=\dfrac{7}{25}\cdot\dfrac{25}{24}=\dfrac{7}{24}$

$Ctg\alpha=\dfrac{1}{tg\alpha } =1:\dfrac{7}{24}=1\cdot\dfrac{24}{7}=3\dfrac{3}{7} \\\\\\Otvet:\boxed{Cos\alpha =\dfrac{24}{25} \ ; \ tg\alpha =\dfrac{7}{24} \ ; \ Ctg\alpha=3\dfrac{3}{7} }$