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Activity 4.5.5. In this activity, we investigate how a sum of two angles identity for the sine function helps us gain a different perspective on the average rate of change of the sine function. Recall that for any function f on an interval [a, a + h], its average rate of changeis MW] 2 W a. Let f(ac) = sin(:v). Use the definition of xii/[01mph] to write an expression for the average rate of change of the sine function on the interval [a + h, a]. b. Apply the sum of two angles identity for the sine function, sin(a l 5) = sin(a) cos(B) l cos(a) 811108 to the expression sin(a + h). c. Explain why your work in (a) and (b) together with some algebra shows that cos(h) 1 h cos( AI/[axhl = 5111(0)) ' d. In calculus, we move from average rate of change to instantaneous rate the sine function on the interval [a l h, a]. b. Apply the sum of two angles identity for the sine function, sin(a + B) : sin(oc) cosw) + cos(a) sin( to the expression Sin(a + h). c. Explain why your work in (a) and (b) together with some algebra shows that cos(h) 1 h cos( AI/[malh] : 8111(0) ' d. In calculus, we move from average rate of change to instantaneous rate of change by letting h approach 0 in the expression for average rate of change. Using a computational device in radian mode, investigate the behavior of (305(k) 1 h as h gets close to 0. What happens? Similarly, how does Sing) behave for small values of 11? What does this tell us about Ell/[awn] for the sine function as h approaches 0