Find the values of x that satisfy sin^2(x) - sin(x) = 0.
Correct Answer: x = 0 and x = π
- Step 1: Start with the equation sin^2(x) - sin(x) = 0.
- Step 2: Notice that this equation can be factored. Rewrite it 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, ... (any integer multiple of π).
- Step 5: Now, set the second factor equal to zero: sin(x) - 1 = 0.
- Step 6: Solve sin(x) - 1 = 0. The solution is sin(x) = 1, which occurs at x = π/2 + 2kπ (where k is any integer).
- Step 7: Combine the solutions from both factors. The values of x that satisfy the original equation are x = 0, π, and π/2 + 2kπ.
- Trigonometric Identities – Understanding the properties of sine and how to manipulate trigonometric equations.
- Factoring Quadratic Equations – Applying factoring techniques to solve equations that can be expressed in a quadratic form.
- Finding Roots – Identifying the values of x that make the equation true by setting each factor to zero.
