Find all solutions
4sin^(2)x - 4sin(x) + 1 = 0
To find all the solutions of the given equation 4sin^2(x) - 4sin(x) + 1 = 0, let's solve it step by step:
Step 1: Simplify the equation if possible.
This equation cannot be simplified any further, so we'll move on to the next step.
Step 2: Recognize the equation as a quadratic equation in terms of sin(x).
The equation can be rewritten as: (2sin(x) - 1)^2 = 0.
Step 3: Solve the quadratic equation.
We have a perfect square on the left side of the equation, which means it can only be equal to zero. Therefore, 2sin(x) - 1 = 0.
Step 4: Solve for sin(x).
Adding 1 to both sides of the equation, we get 2sin(x) = 1. Then, dividing by 2: sin(x) = 1/2.
Step 5: Find the solutions for sin(x) = 1/2.
We know that sin(x) = 1/2 when x is equal to π/6 or 5π/6 (using the unit circle or special angles).
Therefore, the solutions to the equation 4sin^2(x) - 4sin(x) + 1 = 0 are x = π/6 + 2πn and x = 5π/6 + 2πn, where n is any integer.
[2 sin(x) - 1]^2 = 0 ... sin(x) = 1/2
x = 30º + n 360º ... x = 150º + n 360º