Answer:
θ = 0.896 and 2.246.
Step-by-step explanation:
We will use the identity
sin²θ + cos²θ = 1
and the double angle formula
cos2θ = cos²θ - sin²θ = (1 - sin²θ) - sin²θ = 1 - 2sin²θ
1. Convert the equation to a more solvable form.
2. Solve the quadratic
a = 2; b = 1; c = -2
(a) Set up the equation.
(b) Evaluate the roots
3. Solve for θ
(a) The negative root
u = sinθ = -1.281
sinθ cannot be less than -1. No solution.
(b) The positive root
If sin(x) = a, then
x = arcsin(x) + 2πn and
x = π - arcsin(a) + 2πn
(i) Case 1
u = sinθ = 0.7808
θ = arcsin(0.7808) + 2πn
θ = 0.8959 ± 2πn
If n = 0,
θ = 0.8959
(ii) Case 2
θ = π - 0.8959 + 2πn
If n = 0,
θ = π - 0.8959 = 2.246
The solutions are θ = 0.8959 and θ = 2.246
The diagram below shows the graph of f(θ) with its roots at 0.896 and 2.246.
