Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].

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Question: Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].

Options:

Correct Answer: π/6, 5π/6

Solution:

The solutions are x = π/6 and x = 5π/6.

Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].

Find the solutions of the equation 2sin(x) - 1 = 0 in the interval [0, 2π].

Correct Answer: x = π/6 and x = 5π/6

Step 1: Start with the equation 2sin(x) - 1 = 0.
Step 2: Add 1 to both sides of the equation to isolate the sine term: 2sin(x) = 1.
Step 3: Divide both sides by 2 to solve for sin(x): sin(x) = 1/2.
Step 4: Identify the angles where sin(x) equals 1/2. These angles are in the unit circle.
Step 5: The angles that satisfy sin(x) = 1/2 in the interval [0, 2π] are x = π/6 and x = 5π/6.
Step 6: Therefore, the solutions to the equation in the given interval are x = π/6 and x = 5π/6.

Trigonometric Equations – The question tests the ability to solve a basic trigonometric equation involving the sine function.
Interval Restrictions – The solutions must be found within a specified interval, which requires understanding of periodic functions.