Final answer:
The greatest instantaneous rate of change for the graphs of y=sin(x) and y=cos(x) occurs at the midpoints between the peaks and troughs where their derivatives are ±1, specifically at quarter periods for sine (π/2, 3π/2, ...) and at whole periods for cosine (0, π, ...).
Step-by-step explanation:
To determine where on the graphs of y=sin(x) and y=cos(x) the greatest instantaneous rate of change occurs, it is crucial to understand that the instantaneous rate of change of a function at a given point is represented by the slope of the tangent line to the graph at that point. For the functions y=sin(x) and y=cos(x), this corresponds to the derivative of each function.
The derivative of y=sin(x) is y'=cos(x), and the derivative of y=cos(x) is y'=-sin(x). The magnitude of the slope (ignoring the sign) is the greatest when the values of cos(x) and sin(x) are at their maximum absolute value, which is 1. Therefore, the greatest instantaneous rate of change for both graphs occurs when the values of the derivatives are ±1. This happens at the midpoints between the peaks and troughs, where the graphs of sine and cosine intersect the x-axis at their respective quarter periods: π/2, 3π/2, ... for sine and 0, π, ... for cosine.
Thus, the correct answer to the question 'Where on the graphs of y=sin(x) and y=cos(x) would the greatest instantaneous rate of change occur?' is (d) At the midpoints between the peaks and troughs of both graphs.