Question

Se da :cos a=5/13 a€(3π\2;2π) sa se calculeze ctg2a

Answer 1

[tex]\sin^2 a+\cos^2 a=1\\ sin^2 a+\dfrac{25}{169}=1\\ sin^2 a=1-\dfrac{25}{169}\\ sin^2 a=\dfrac{144}{169}\\ a\in \left(\dfrac{3 \pi}{2}, 2 \pi\right)\Rightarrow \sin a=-\dfrac{12}{13}\\ ctg\ 2a=\dfrac{\cos 2a}{\sin 2a}=\dfrac{2\cdot cos^2 a-1}{2\sin a\cos a}=\dfrac{ 2\cdot \frac{25}{169}-1}{2\cdot (-\frac{12}{13}) \cdot \frac{5}{13}}=\dfrac{\frac{50-169}{169}}{-\frac{120}{169}}=\boxed{\dfrac{119}{120}}[/tex]