Question

3Sin x + 4cos x +5 sin 3x=0

Answer 1

$3sin x+4cos x+5sin3x=0\ \ 5sin(x+ frac{pi}{4} )+5sin 3x=0\ \ sin (x+frac{pi}{4} )+sin 3x=0\ \ 2sin frac{x+frac{pi}{4} +3x}{2} cos frac{x+frac{pi}{4} -3x}{2} =0\ \ 2sin(x+frac{pi}{8} )cos(frac{pi}{8} -x)=0\ \ left[begin{array}{ccc}sin (x+frac{pi}{8})=0 \ cos (frac{pi}{8} -x)=0end{array}right~~~Rightarrow~~~~ boxed{ left[begin{array}{ccc}x_1=-frac{pi}{8} + pi k,k in mathbb{Z}\ x_2=frac{pi}{8} -2 pi k,k in mathbb{Z}end{array}right}$