Sin θ − 4 = 8 sin θ?

Question

Solve for all degree solutions and if 0° ≤ θ < 360°.

Answer 1

  

The trigonometric equation is sin θ - 4 = 8sin θ.

Subtract 8sin θ from each side.

sin θ - 4 - 8sin θ = 0

- 7sin θ - 4 = 0

Add 4 to each side.

- 7sin θ = 4

Divide each side by negative 7.

sin θ = - 4/7.

sin θ = sin[sin^{- 1}(- 4/7)]

sin θ = sin[- 34.85⁰].

The genaral solution of sin(θ) = sin(α) is θ = nπ + (- 1)ⁿα, where n is an integer.

⇒ θ = nπ + (- 1)ⁿ(- 34.85⁰)

If n = 0, θ = (0)π + (- 1)⁰(- 34.85⁰) = - 34.85⁰.

If n = 1, θ = (1)π + (- 1)¹(- 34.85⁰) =180⁰ + (- 1)¹(- 34.85⁰) = 180⁰ + 34.85⁰ = 214.84⁰

If n = 2, θ = (2)π + (- 1)²(- 34.85⁰) =360⁰ - 34.85⁰ = 325.15⁰.

The solutions of the given equation are θ = 214.84⁰ and θ = 325.15⁰ in the interval [0⁰, 360⁰).