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

The general solution for sin(2x)cos(x)+sin(x)=0 is x=2pin,x=pi+2pin

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

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

x = 2 πn, x = π + 2 πn

Solution steps

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

Rewrite using trig identities

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

Factor

sin (x) + 2 cos^{2} (x) sin (x) : sin (x) (2 cos^{2} (x) + 1)

sin (x) (2 cos^{2} (x) + 1) = 0

Solving each part separately

sin (x) = 0 or 2 cos^{2} (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos^{2} (x) + 1 = 0 :

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

cos (2 x) + cos (x) = 1 + sin (2 x) - sin (x)

- sin (θ) + 2 sin^{2} (θ) = 0

- sec (x) + 2 sec (x) cos (x) = 0

2 tan (x) - 3 sin (x) = 0

sin (x) + 5 = 4

What is the general solution for sin(2x)cos(x)+sin(x)=0 ?
The general solution for sin(2x)cos(x)+sin(x)=0 is x=2pin,x=pi+2pin