Question

what is the solution to sinx = sin3x

Answer 1

same solution as
x = 3x

x = 0

Note for any multiple of pi, the sin is also 0
solution is 0, pi, 2pi ... npi

Answer 2

wait that is only half the solution, sorry to get full solution you need expand sin(3x)

http://en.wikipedia.org/wiki/Trig_identity#Double-angle.2C_triple-angle.2C_and_half-angle_formulae

now you have
\[\sin x = -4\sin^3 x +3\sin x\]
which factors to
\[2 \sin x(2 \sin^2 x -1) = 0\]
which gives solutions
\[x = n \pi , \frac{\pi}{4} + n \frac{\pi}{2}\]

Answer 3

Alternatively use
\[\sin A -\sin B =2\cos (\frac{ A+B }{2 })\sin (\frac{ A-B }{2 })\]

Answer 4

\[\sin3x=sinx \\3x=x+2k \pi\\3x=\pi-x+2k \pi\\3x=x+2k \pi \rightarrow 2x=2k \pi \rightarrow x=k \pi\\3x=\pi-x+2k \pi \rightarrow 4x=\pi+2k \pi \rightarrow x=\frac{ \pi }{ 4}+\frac{ k \pi }{ 2 }\]

Answer 5

\[k \in Z\]