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

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

Options:

Correct Answer: x = nπ

Solution:

Using the identity sin(a) = sin(b) gives x = nπ or x = (2n+1)π/3.

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

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.