Final answer:
The equation sin(θ) = √2/2 is solved for all solutions by finding the angles where the sine function equals √2/2 on the unit circle, which are θ = π/4 and θ = 3π/4, and then adding multiples of the period 2π for each solution.
Step-by-step explanation:
To solve the equation sin(θ) = √2/2 for all solutions, we need to consider the unit circle where the sine function has the value of √2/2. This occurs at two principal angles in the unit circle, which are θ = π/4 and θ = 3π/4. Since sine has a period of 2π, the general solutions for θ where sin(θ) equals √2/2 are given by adding integer multiples of 2π to these principal angles.
The correct answers for the equation sin(θ) = √2/2 are therefore:
- θ = π/4 + 2nπ
- θ = 3π/4 + 2nπ
Where n is an arbitrary integer. This allows us to include all possible solutions that lie in different cycles of the sine function's periodic behavior.