Solve

Question

cos(2theta)+8 sin^2(theta)=4?

Answer 1

The trigonometric equation is cos(2θ) + 8sin² (θ) = 4.

Double angle formulas : cos(2θ) = 1 - 2sin² θ.

1 - 2sin² θ + 8sin² (θ) = 4

1 + 6sin² (θ) = 4

6sin² (θ) = 3

sin² (θ) = 3/6 = 1/2

⇒ sin (θ) = ± 1/√2

sin (θ) = - 1/√2 and sin (θ) = + 1/√2.

• sin (θ) = - 1/√2.

sin(θ) = sin(3π/4)

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

⇒ θ = nπ + (- 1)ⁿ(3π/4).

• sin (θ) = 1/√2.

sin(θ) = sin(π/4)

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

⇒ θ = nπ + (- 1)ⁿ(π/4).

The solutions of the given equation are θ = nπ + (- 1)ⁿ(3π/4) and θ = nπ + (- 1)ⁿ(π/4), where n is an integer.