Final answer:
To solve the equation cos(π/2-x) = √3/2 within the interval [0,2π], we convert cosine to sine and find that x equals π/3 and 2π/3, as sine reaches √3/2 at these angles.
Step-by-step explanation:
To find a solution to the equation cos(π/2-x) = √3/2 for all possible values of x on the interval [0,2π], we use the identity cos(θ) = sin(π/2 - θ), which gives us sin(x) = √3/2. The values of x that satisfy this equation occur twice in the interval [0,2π], once in the first quadrant and once in the second quadrant, where the sine function is positive.
The sine of an angle is √3/2 at 60° (π/3 radians) and 120° (2π/3 radians). Therefore, the solutions for x in the interval [0,2π] are x = π/3 and x = 2π/3.