Question

Use an Addition or Subtraction Formula to simplify the equation. sin(3θ) cos(θ) − cos(3θ) sin(θ) = Square root 2/2 Find all solutions in the interval [0, 2π). (Enter your answers as a comma-separated list.)

Answer 1

`\sin3\theta\cos\theta-\cos3\theta\sin\theta=\sin(3\theta-\theta)=\sin2\theta=\frac{\sqrt2}2`

`\sin2\theta=\frac1{\sqrt2}`

`\implies2\theta=\frac\pi4+2n\pi,\,2\theta=\frac{3\pi}4+2n\pi`

`\implies\theta=\frac\pi8+n\pi,\,\theta=\frac{3\pi}8+n\pi`

where
`n` is any integer. To take only the solutions within the interval
`0\le\theta<2\pi`, we solve


`0\le\frac\pi8+n\pi<2\pi\implies\frac18+n<2\implies n<\frac{15}8\implies n=0,\,n=1`

`\implies\theta=\frac\pi8,\,\theta=\frac\pi8+\pi=\frac{9\pi}8`


`0\le\frac{3\pi}8+n\pi<2\pi\implies \frac38+n<2\implies n<\frac{13}8\implies n=0,\,n=1`

`\implies\theta=\frac{3\pi}8,\,\theta=\frac{11\pi}8`

Answer 2

Answer: For 0 ≤Ф≥ 2π (where π= 180°)

∴ Ф = 22.5°, 67.5°, 112.5°, 157.5°, 202.5°, 247.5°, 292.5°, 337.5°

Step-by-step explanation:

sin(3Ф)cos(Ф) - cos(3Ф)sin(Ф) = √2/2

sin(3Ф - Ф) =√2/2

3Ф -Ф = sin∧-1{√2/2}

2Ф = 45°

∴ Ф = 22.5°