What is the general solution for sqrt(2)sin(x)+sin(2x)=0 ?

The general solution for sqrt(2)sin(x)+sin(2x)=0 is x=2pin,x=pi+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin

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sqrt(2)sin(x)+sin(2x)=0

2 sin (x) + sin (2 x) = 0

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Solution steps

2 sin (x) + sin (2 x) = 0

Rewrite using trig identities

sin (x) 2 + 2 cos (x) sin (x) = 0

Factor

sin (x) 2 + 2 cos (x) sin (x) : sin (x) (2 + 2 cos (x))

sin (x) (2 + 2 cos (x)) = 0

Solving each part separately

sin (x) = 0 or 2 + 2 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 + 2 cos (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

2 sin^{2} (x) + sin (x) = 0

tan^{2} (3 x) = 3

cos (x) - 3 sin (x) = 2

tan (x) = - 3

sec (θ) = - 2

What is the general solution for sqrt(2)sin(x)+sin(2x)=0 ?
The general solution for sqrt(2)sin(x)+sin(2x)=0 is x=2pin,x=pi+2pin,x=(3pi)/4+2pin,x=(5pi)/4+2pin