Final answer:
The equation sin(3x - π/4) = -1 has 4 roots within the range -2020 to 2020.
Step-by-step explanation:
The equation sin(3x - π/4) = -1 represents a sine function with an amplitude of 1 and a period of 2π/3. The range of -2020 to 2020 is equivalent to -2π/3 to 2π/3 in terms of the argument 3x - π/4.
To determine the number of roots within this range, we need to find the number of full cycles that the sine function completes within the range. Since each full cycle has 2 roots, the number of roots within the range is equal to the number of full cycles.
Dividing the total range 4π/3 by the period 2π/3, we find that there are 2 full cycles within the range. Therefore, the equation has a total of 4 roots within the range -2020 to 2020.