Hi!
In this problem I need to find all real numbers x.
$\displaystyle 5sin(x)+cos(2x)=3$
and
$\displaystyle 5sin(x)+cos(2x)=6$
Can you help me?
Recall that $\displaystyle \cos(2x)=\cos^2x-\sin^2x=1-2\sin^2x$.
Thus, $\displaystyle 5\sin x+\cos(2x)=3\implies 5\sin x+1-2\sin^2 x=3\implies2\sin^2 x-5\sin x+2=0$
Let $\displaystyle u=\sin x$. Then solve $\displaystyle 2u^2-5u+2=0$, and then find $\displaystyle x$.
Do something similar for the second one.