Final answer:
The solution to the trigonometric inequality sin(x) > cos(x) over the interval 0 ≤ x ≤ 2π radians is π/4 ≤ x ≤ 3π/4. The inequality is based on the fact that sin(x) = cos(π/2 - x), and within the specified range, the cosine function decreases, making sin(x) greater than cos(x).
Step-by-step explanation:
To solve the trigonometric inequality sin(x) > cos(x) over the interval 0 ≤ x ≤ 2π radians, we first note that both sine and cosine functions range between -1 and 1. To find when sin(x) is greater than cos(x), we can look at the unit circle or use trigonometric identities.
One approach is to use the fact that sin(x) = cos(π/2 - x), which means the inequality sin(x) > cos(x) is equivalent to cos(π/2 - x) > cos(x). This inequality holds true when the angle x is in the interval (π/4, 3π/4) because in this range, the cosine function is decreasing, thus making the cos(x) value smaller than that of cos(π/2 - x).
Therefore, the correct solution for when sin(x) is greater than cos(x) in the given interval is π/4 ≤ x ≤ 3π/4, which corresponds to answer choice C).