Find the general solution for each of the following equation: `sin x + sin 3x + sin 5x = 0`

Explanation

We have, `sin x+sin 3x+sinn 5x=0`
`therefore (sin x+sin 5x) +sin 3x=0`
`therefore 2 sin ((x+5x)/(2))cos ((5x-x)/(2))+sin3x=0`
`therefore2 sin 3x cos 2x+sin 3x=0`
`therefore sin3x(2 cos2x+1) =0`
Either `sin 3x=0 or 2 cos 2x+1=0`
i.e., `sin 3x=0 or cos 2x=-1/2`
Now, `cos 2x=-cos ""(pi)/(2)`
`cos 2x=cos(pi-(pi)/(3))`
`cos 2x=cos ""(2pi)/(3)`
`therefore sin 3x=0or cos 2x=cos ""(2pi)/(3)`
`3pi=npi, n ne Zor 2x=2mpipm(2pi)/(3)` where` m inZ`
Hence, `x=(npi)/(3)or x=mpipm (pi)/(3), wheren, m in Z`