Find the general solution of the equation sin(x) = sin(2x).
Correct Answer: x = nπ or x = (2n+1)π/3
- Step 1: Start with the equation sin(x) = sin(2x).
- Step 2: Use the identity sin(a) = sin(b), which means a = b + 2nπ or a = π - b + 2nπ for any integer n.
- Step 3: Set up the first case: x = 2x + 2nπ.
- Step 4: Rearrange the first case to find x: x - 2x = 2nπ, which simplifies to -x = 2nπ, so x = -2nπ.
- Step 5: Set up the second case: x = π - 2x + 2nπ.
- Step 6: Rearrange the second case to find x: x + 2x = π + 2nπ, which simplifies to 3x = π + 2nπ, so x = (π + 2nπ) / 3.
- Step 7: Combine the solutions from both cases: x = -2nπ and x = (π + 2nπ) / 3.
- Step 8: Notice that (π + 2nπ) / 3 can be rewritten as (2n + 1)π / 3 for clarity.
- Trigonometric Identities – Understanding the identity sin(a) = sin(b) and how to apply it to find solutions.
- General Solutions of Trigonometric Equations – Finding all possible solutions for the equation within the context of periodic functions.
