Final answer:
The equation cos(x) = √3/2 is solved within the interval [0,2π], yielding the angles x = π/6 and x = 11π/6 as the solutions.
Step-by-step explanation:
To solve the equation cos(x) = √3/2 in the interval [0,2π], we first need to identify the angles where the cosine value is √3/2. The cosine function has this value at angles of π/6 (30°) and 11π/6 (330°) within one full circle (0 to 2π). Since cosine is positive in the first and fourth quadrants, these are the only solutions in the given interval.
Here's a step-by-step process to find these solutions:
- Looking at the unit circle, we identify where the cosine value is √3/2.
- We find that this occurs at angles π/6 and 11π/6.
- Check that both solutions are within the given interval [0,2π].
Therefore, the solutions to the equation cos(x) = √3/2 in the interval [0,2π] are x = π/6 and x = 11π/6.