precal question

Question

Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2𝜋). (Enter your answers as a comma-separated list.)

tan𝜃/2) − sin(𝜃) = 0

𝜃 = 

 

 

Answer 1

tan(θ/2) -sin(θ) = 0

=> sin(θ/2)/cos(θ/2) - 2sin(θ/2)cos(θ/2) = 0

=> sin(θ/2) -2sin(θ/2)cos²(θ/2) = 0, cos(θ/2)≠ 0

=> sin(θ/2) (1- 2cos²(θ/2))=0

This gives you three equations:

sin(θ/2) = 0 or 1-√2 cos(θ/2) = 0 or 1+√2 cos(θ/2) = 0

=>θ/2 = n pi

=> θ = 0 , as θ is in [0, 2𝜋)

1-√2 cos(θ/2) = 0 => cos(θ/2)= 1/√2 = cos(𝜋/4)

=> θ/2 =𝜋/4, 7𝜋/4,..

θ = 𝜋/2

1+√2 cos(θ/2) = 0 => cos(θ/2)= -1/√2 = cos(3𝜋/4)

θ/2= 3𝜋/4

θ = 3𝜋/2

θ = 0, 𝜋/2, 3𝜋/2 Ans