What are the solutions of the equation cos(x) = -1/2 in the interval [0, 2π]?

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Q: What are the solutions of the equation cos(x) = -1/2 in the interval [0, 2π]?

Solution: The solutions are x = 2π/3 and 4π/3.

Steps: 7

Step 1: Understand the equation cos(x) = -1/2. This means we are looking for angles x where the cosine value is -1/2.
Step 2: Recall the unit circle and the values of cosine for common angles. Cosine is negative in the second and third quadrants.
Step 3: Identify the reference angle where cos(x) = 1/2. This angle is π/3 (or 60 degrees).
Step 4: Since we need cos(x) = -1/2, we find the angles in the second and third quadrants. The angles are: π - π/3 for the second quadrant and π + π/3 for the third quadrant.
Step 5: Calculate the angles: In the second quadrant, x = π - π/3 = 2π/3. In the third quadrant, x = π + π/3 = 4π/3.
Step 6: List the solutions found: The solutions are x = 2π/3 and x = 4π/3.
Step 7: Ensure both solutions are within the interval [0, 2π]. Both 2π/3 and 4π/3 are within this interval.