Question

Solve:

sin(2x)=sinx

Answer 1

x=2npi n is an integer

Answer 2

2x=x

Answer 3

2x-x=0
1x=0
x=0

Answer 4

double angle formula for sin2x:
sin2x=sinx
2sinxcosx=sinx
sinx=0 or cosx=1/2
x=0,2pi,pi/3,5pi/3

Answer 5

largrange includes other solutions
so he is right

Answer 6

we have 2sinxcosx=sinx
2sinxcosx-sinx=0
sinx(2cosx-1)=0

Answer 7

sinx=0 or 2cosx-1=0

Answer 8

myininaya is also right, her answer is very thorough

Answer 9

\[\sin(2x)=2\sin(x)\cos(x)\]
\[\sin(2x)=\sin(x) \iff 2\sin(x)\cos(x)=\sin(x) \iff 2\cos(x)=1 \iff \cos(x)=\frac{1}{2}\]
\[\iff x=\frac{\pi}{3} + 2k \pi, k \in \mathbb{Z} \ or x=-\frac{\pi}{3}+2k \pi, k \in \mathbb{Z}\]

Answer 10

x=2npi, n is an integer
cosx=1/2 when x=pi/3+2npi and x=5pi/3+2npi