Find the general solution of the equation sin(x) + sin(2x) = 0.

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Question: Find the general solution of the equation sin(x) + sin(2x) = 0.

Options:

Correct Answer: x = nπ

Solution:

Factoring gives sin(x)(1 + 2cos(x)) = 0, leading to x = nπ or cos(x) = -1/2.

Find the general solution of the equation sin(x) + sin(2x) = 0.

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.