Find the general solution of the equation sin(x) + sin(2x) = 0.
Correct Answer: x = nπ or x = (2n + 1)π/3, n ∈ Z
- Step 1: Start with the equation sin(x) + sin(2x) = 0.
- Step 2: Use the double angle identity for sine: sin(2x) = 2sin(x)cos(x).
- Step 3: Substitute this identity into the equation: sin(x) + 2sin(x)cos(x) = 0.
- Step 4: Factor out sin(x) from the equation: sin(x)(1 + 2cos(x)) = 0.
- Step 5: Set each factor equal to zero: sin(x) = 0 and 1 + 2cos(x) = 0.
- Step 6: Solve sin(x) = 0. The solutions are x = nπ, where n is any integer.
- Step 7: Solve 1 + 2cos(x) = 0. Rearranging gives cos(x) = -1/2.
- Step 8: Find the solutions for cos(x) = -1/2. The solutions are x = (2n + 1)π/3, where n is any integer.
- Step 9: Combine all solutions: x = nπ and x = (2n + 1)π/3.
- Trigonometric Identities – Understanding and applying the properties of sine and cosine functions.
- Factoring Techniques – Using factoring to simplify trigonometric equations.
- General Solutions of Trigonometric Equations – Finding all possible solutions for trigonometric equations over the set of real numbers.
