Use identities to write the equation cos2A=2sin^2A in terms of the single angle .

Question

Then solve the equation for 0 <= A < 2pi.

Answer 1

Equation : cos(2A) = 2sin²(A).

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

1 - 2sin²(A) = 2sin²(A)

1 = 4sin²(A)

1/4 = sin²(A)

sin(A) = ± 1/2.

sin(A) = 1/2 and sin(A) = - 1/2.

sin(A) = sin(π/6) and sin(A) = - sin(π/6) = sin(- π/6).

General solution : If sin(θ) = sin(α) then θ = nπ + (-1)ⁿ α, where n is integer.

If α = π/6 then angle A = nπ + (-1)ⁿ (π/6).

If α = - π/6 then angle A = nπ + (-1)ⁿ (- π/6) = nπ - (-1)ⁿ (- π/6).

The solutions in the interval (0, 2π) are π/6, 5π/6 7π/6 and 11π/6.