Final answer:
The equation sin(100x) = √3/2 has 100 solutions between [0, 2π). The smallest solution is x = π/300, and the largest solution is x = 2π - π/300.
Step-by-step explanation:
The student is asking to solve the equation sin(100x) = √3/2 to determine the number of solutions between [0, 2π), the smallest solution, and the largest solution in this interval. We know that sin(θ) = √3/2 at θ = π/3 and 2π/3 in the interval [0, 2π) because these are the angles in the first and second quadrants where the sine of an angle is positive and equal to √3/2.
For the general solution sin(α) = sin(β), the solutions are α = n(2π) + (-1)^nβ where n is an integer. Therefore, the solutions for sin(100x) = √3/2 are x = π/300 + k(2π/100) and x = 2π/300 + k(2π/100), with k being an integer.
To find the number of solutions in [0, 2π), we divide the interval length, 2π, by the period of sin(100x), which is 2π/100, getting 100 possible solutions. However, since we are looking for x values and not angles in radians multiplied by 100, we'll have two solutions for each k from 0 to 49, providing 100 total solutions between [0, 2π).
The smallest solution is when k = 0, which gives us x = π/300. The largest solution in [0, 2π) is when k = 49, for the second set, which gives us x = (2π/300) + 49(2π/100) = 2π - π/300.