Question

Find the general solution of sin x + sin 2x + sin 3x + sin 4x = 0 ?

Answer 1

Have you tried using identities to solve it?

Answer 2

Like sin(2x)=2sin(x)cos(x)

Answer 3

no

Answer 4

For sin 3x , split it then use identity
=sin(3x)
=sin(2x+x)

Use sin(A+B)=sin(A)cos(B)+cos(A)sin(B)

Answer 5

sinx+sin2x+sin2xcosx+cos2xsinx+sin4x

Answer 6

i have put the value in the given question

Answer 7

as for sin4x as well, split it sin(2x+2x)

Answer 8

then use sin(2x)=2sin(x)cos(x) for all

Answer 9

so the given equation will becomes
sinx+2sinxcosx+sin2xcosx+cos2xsinx+sin2xcos2x+cos2xsin2x

Answer 10

yes then use sin(2x)=2sin(x)cos(x) for those with sin(2x)

Answer 11

You should get something like this
sin(x) + 2sin(x)cos(x) + [2sin(x)cos^2(x)+sin(x)cos(2x)] + 4sin(x)cos(x)cos(2x) = 0

Answer 12

Then factor sin(x) out and you will get a solution

Answer 13

sinx+2sinxcosx+2sinxcosxcosx+cos2xsinx+2sinxcosxcos2x+2cos2xsinxcosx=0

Answer 14

Try simplify it you should get something like this
sin(x) + 2sin(x)cos(x) + [2sin(x)cos^2(x)+sin(x)cos(2x)] + 4sin(x)cos(x)cos(2x) = 0

Answer 15

Once you've done it then factor sin(x),
sin(x)[1 + 2cos(x) + 2cos^2(x) + cos(2x) + 4cos(x)cos(2x)] = 0

You can say from here, sin(x)=0 then x = 2nπ for any integer n

Answer 16

so x= 2nπ
is the final answer