Final answer:
To solve the equation 1-sinθ = 2cos²θ, we need to manipulate the equation to get it in terms of one trigonometric function. First, we use the identity 1 - cos²θ = sin²θ. We find that sinθ = -1/2 or sinθ = 1, and using the unit circle or trigonometric identities, we can determine the values of θ.
Step-by-step explanation:
To solve the equation 1 - sinθ = 2cos²θ, we need to manipulate the equation to get it in terms of one trigonometric function. First, we can use the identity 1 - cos²θ = sin²θ. So, we can rewrite the equation as 1 - sinθ = 2(1 - sin²θ). Expanding and simplifying, we get 1 - sinθ = 2 - 2sin²θ. Rearranging the terms, we have 2sin²θ - sinθ - 1 = 0. Now, we can solve this quadratic equation for sinθ.
Using factoring or quadratic formula, we find that sinθ = -1/2 or sinθ = 1. To find the values of θ, we can use the unit circle or trigonometric identities. For the first equation, sinθ = -1/2, the solutions are θ = 210° + 360°n and θ = 330° + 360°n, where n is an integer. For the second equation, sinθ = 1, the solution is θ = 90°.