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

The general solution for 2sin(2x)=sin(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin

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2sin(2x)=sin(x)

2 sin (2 x) = sin (x)

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Solution steps

2 sin (2 x) = sin (x)

Subtract

sin (x)

from both sides

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

Rewrite using trig identities

- sin (x) + 4 cos (x) sin (x) = 0

Factor

- sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) - 1)

sin (x) (4 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 4 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

sin (\frac{11 π}{4})

arccos (\frac{4}{5})

cos (θ) = 1

1 + sin (θ) = 2 cos^{2} (θ)

arccsc (- 2)

What is the general solution for 2sin(2x)=sin(x) ?
The general solution for 2sin(2x)=sin(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin