Question

Calculati cu ajutorul formulelor trigonometrice: e) si f)

Answer 1

$\it e)\ \dfrac{tg80^o-tg5^o}{1+tg80^otg5^o}=tg(80^o-5^o)=tg75^o=tg(30^o+45^o)=\\ \\ \\ =\dfrac{tg30^o+tg45^o}{1-tg30^otg45^o}=\dfrac{\dfrac{\sqrt3}{3}+1}{1-\dfrac{\sqrt3}{3}\cdot1}=\dfrac{3+\sqrt3}{3-\sqrt3}=\dfrac{(3+\sqrt3)^2}{9-3}=\\ \\ \\ =\dfrac{9+6\sqrt3+3}{6}=\dfrac{12+6\sqrt3}{6}=\dfrac{6(2+\sqrt3)}{6}=2+\sqrt3$

$f)\ cos68^ocos78^o+cos22^ocos12^o-cod19^o=\\ \\ \\ =\dfrac{1}{2}[cos(68^o+78^o)+cos(78^o-68^o)+cos(22^o+12^o)+cos(22^o-12^o)]-cos10^o=\\ \\ \\ =\dfrac{1}{2}(cos146^o+2cos10^o+cos34^o)-cos10^o=\dfrac{cos146^o+cos34^o}{2}=\\ \\ \\ =\dfrac{2cos\dfrac{146^o+34^o}{2}cos\dfrac{146^o-34^o}{2}}{2}=cos90^ocos112^o=0\cdot cos112^o=0$