Question

Find the general solution of `sin x+sin3x+sin5x=0`

Answer 1

`sinx+sin3x+sin5x=0`

`therefore(sinx+sin5x)+sin3x=0`

`therefore 2sin((x+5x)/2)cos((5x-x)/2)+sin3x=0`

∴ `2sin3xcos2x+sin3x=0`

∴ `(2cos2x+1)sin3x=0`

∴ sin 3x = 0      or    2 cos 2x + 1 = 0

∴ sin 3x = 0 ...(i)   or  cos 2x = `-1/2`  ....(ii)

For (ii) cos 2x = - cos `pi/3`

`therefore "cos 2x" = "cos" (pi - pi/3)`  .....(by allied angles)

cos 2x = cos `(2pi)/3`

∴ from (i) and (ii) we get

∴ sin 3x = 0         or            cos 2x = cos `(2pi)/3`

∴ 3x = n π , n ∈ Z       or     2x = 2mx `+- (2pi)/3 , "where"   "m" ∈ "Z".`

Here the required solution is

∴ `"x" = ("n"pi)/3`  or  `"x" ="m"pi +- pi/3 , "where" "n", "m" ∈ "Z".`