Question

Ayuda!!
si:
sen(x+10°) = cos(x+40°)
hallar:
tg3x + (4√3) (sen(x+10°))

Answer 1

$\sin (x+10)=\cos(x+40)\\ \\ \sin (x+10)=\sin(50-x)\\ \\ \sin (x+10)-\sin(50-x)=0\\ \\ 2\cos\left(\dfrac{x+10+50-x}{2}\right)\sin\left(\dfrac{x+10-50+x}{2}\right)=0\\ \\ \\ \cos(30)\sin(x-20)=0\\ \\ \sin(x-20)=0\\ \\ x-20=k\pi\,,\, \forall k\in \mathbb Z\\ \\ \boxed{x=20+k\pi}$

Entonces

$E=\tan3x+4\sqrt3\sin(x+10)\\ \\ E=\tan[3(20+k\pi)]+4\sqrt3\sin(20+k\pi+10)\\ \\ E=\tan(60+3k\pi)+4\sqrt3\sin(30+k\pi)\\ \\ E=(-1)^{k}\sqrt3+4\sqrt3\sin(30)\\ \\ \boxed{E=(-1)^k\sqrt3+2\sqrt3}$