Find the exact solutions to sin(4x)cos(x)=cos(4x)sin(x) in the interval [0,2π). If the equation has no solutions, respond with DNE.

Question

Find the exact solutions to sin(4x)cos(x)=cos(4x)sin(x) in the interval [0,2π). If the equation has no solutions, respond with DNE.

Answer 1

have

sin(4x)cos(x)-cos(4x)sin(x)=0

same as

sin(3x)=0,,,,sin(4x-x)=sin(3x)

has six solutions x=0,pi/3,2*pi/3,pi,4*pi/3,5*pi/3

3x=0+2k*pi

x=0+2k*pi/3,,,,k=0,1,2

and

3x=pi+2k*pi

x=pi/3+2k*pi/3,,,k=0,1,2

Answer 2

Algebraic manipulation: Divide both sides by cos(4x) and cos(x). Convert sin/cos to tan. Now, solve the values of x that makes both sides equal.