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

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

Options:

Correct Answer: -1, 2

Solution:

Factoring gives (sin(x) - 2)(sin(x) + 1) = 0, so sin(x) = 2 (not possible) or sin(x) = -1, giving x = 3π/2.

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

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

Step 1: Start with the equation sin^2(x) - sin(x) - 2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of sin(x).
Step 3: Rewrite the equation as (sin(x))^2 - sin(x) - 2 = 0.
Step 4: Factor the quadratic equation. Look for two numbers that multiply to -2 and add to -1.
Step 5: The numbers -2 and 1 work, so we can factor it as (sin(x) - 2)(sin(x) + 1) = 0.
Step 6: Set each factor equal to zero: sin(x) - 2 = 0 and sin(x) + 1 = 0.
Step 7: Solve the first equation: sin(x) - 2 = 0 gives sin(x) = 2. This is not possible because the sine function only ranges from -1 to 1.
Step 8: Solve the second equation: sin(x) + 1 = 0 gives sin(x) = -1.
Step 9: Find the values of x where sin(x) = -1. This occurs at x = 3π/2 + 2kπ, where k is any integer.

Trigonometric Equations – The question tests the ability to solve equations involving trigonometric functions, specifically sine.
Factoring Quadratics – The question requires factoring a quadratic expression to find solutions.
Understanding Range of Sine Function – The question assesses knowledge of the range of the sine function, which is limited to [-1, 1].