What are the solutions of the equation sin^2(x) - sin(x) = 0?

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Question: What are the solutions of the equation sin^2(x) - sin(x) = 0?

Options:

Correct Answer: x = nπ

Solution:

Factoring gives sin(x)(sin(x) - 1) = 0, so sin(x) = 0 or sin(x) = 1. Thus, x = nπ or x = π/2 + 2nπ.

What are the solutions of the equation sin^2(x) - sin(x) = 0?

What are the solutions of the equation sin^2(x) - sin(x) = 0?

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 = nπ, where n is any integer.
Step 5: Now, set the second factor equal to zero: sin(x) - 1 = 0.
Step 6: Solve sin(x) - 1 = 0. This gives sin(x) = 1.
Step 7: The solutions for sin(x) = 1 are x = π/2 + 2nπ, where n is any integer.
Step 8: Combine the solutions from both factors. The final solutions are x = nπ or x = π/2 + 2nπ.

Trigonometric Equations – The question tests the ability to solve equations involving trigonometric functions, specifically sine.
Factoring – The solution requires factoring a quadratic expression in terms of sine.
General Solutions – The question assesses understanding of how to express solutions in terms of general forms involving integers.