Determine the values of x that satisfy the equation sin^2(x) - sin(x) = 0.

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Question: Determine the values of x that satisfy the equation sin^2(x) - sin(x) = 0.

Options:

Correct Answer: 0, π/2, π, 3π/2

Solution:

Factoring gives sin(x)(sin(x) - 1) = 0, so x = 0, π/2, π, 3π/2.

Determine the values of x that satisfy the equation sin^2(x) - sin(x) = 0.

Determine the values of x that satisfy the equation sin^2(x) - sin(x) = 0.

Correct Answer: 0, π/2, π, 3π/2

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 = 0, π, 2π, ... (all multiples of π).
Step 5: Now, set the second factor equal to zero: sin(x) - 1 = 0.
Step 6: Solve sin(x) - 1 = 0. The solution is sin(x) = 1, which gives x = π/2 + 2kπ (where k is any integer).
Step 7: Combine the solutions from both factors. The specific solutions in the range [0, 2π) are x = 0, π/2, π, 3π/2.

Trigonometric Identities – Understanding the properties of sine and how to manipulate trigonometric equations.
Factoring Quadratic Equations – Applying factoring techniques to solve equations in the form of a product equal to zero.
Finding Solutions in Trigonometric Functions – Identifying all possible angles that satisfy the given trigonometric equation within a specified interval.