Determine the values of x that satisfy sin^2(x) - sin(x) = 0.
Correct Answer: x = 0, π/2, π, 3π/2
- Step 1: Start with the equation sin^2(x) - sin(x) = 0.
- Step 2: Factor the equation. This can be rewritten as sin(x)(sin(x) - 1) = 0.
- Step 3: Set each factor equal to zero. First, set sin(x) = 0.
- Step 4: Solve sin(x) = 0. The solutions are x = 0, π, 2π, ... (but we will focus on the first cycle).
- Step 5: Now, set the second factor equal to zero: sin(x) - 1 = 0.
- Step 6: Solve sin(x) - 1 = 0. This gives sin(x) = 1.
- Step 7: The solution for sin(x) = 1 is x = π/2 + 2kπ, where k is any integer. For the first cycle, we take k = 0, so x = π/2.
- Step 8: Combine all the solutions from both factors. The solutions in the first cycle are x = 0, π/2, π, and 3π/2.
- Trigonometric Identities – Understanding the properties of sine and how to manipulate trigonometric equations.
- Factoring Quadratic Equations – Recognizing and solving quadratic-like equations in trigonometric contexts.
- Unit Circle – Identifying angles and their sine values on the unit circle.
