Trig Identities

Question

sin(x) = -sin(2x) + sin(5x)

 

How do I solve for ALL solutions?

Answer 1

sin(x)=-sin(2x)+sin(5x)

rearrange the terms

sin(5x)-sin(x)=sin(2x)

sin(a)-sin(b)=2cos([a+b]/2)sin([a-b]/2)

therefore the left side becomes...

2cos([5x+x]/2)sin([5x-x])/2)

2cos(3x)sin(2x)

now we have...

2cos(3x)sin(2x)=sin(2x)

2cos(3x)sin(2x)-sin(2x)=0

factor out sin(2x)

sin(2x)(2cos[3x]-1)=0

sin2x=0 or 2cos(3x)-1=0

 

sin(2x)=0

2x=0º or 2x=180º on the unit circle

x=0º or x=90º

 

2cos(3x)-1=0

2cos(3x)=1

cos(3x)=1/2

3x=60º or 3x=300º on the unit circle

x=20º or x=100º

all solutions: x=0º, x=90º, x=20º, x=100º