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 \([a,a+h]\text{,}\) its average rate of change is
\begin{equation*}
AV_{[a,a+h]} = \frac{f(a+h)-f(a)}{h}\text{.}
\end{equation*}
(a)
Let \(f(x) = \sin(x)\text{.}\) Use the definition of \(AV_{[a,a+h]}\) to write an expression for the average rate of change of the sine function on the interval \([a,a+h]\text{.}\)
(b)
Apply the sum of two angles identity for the sine function, \(\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)\text{,}\) to the expression \(\sin(a+h)\text{.}\)
(c)
Explain why your work in (a) and (b) together with some algebra shows that
\begin{equation*}
AV_{[a,a+h]} = \sin(a) \cdot \frac{\cos(h)-1}{h} - \cos(a) \cdot \frac{\sin(h)}{h}\text{.}
\end{equation*}
(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
\begin{equation*}
\frac{\cos(h)-1}{h}
\end{equation*}
as \(h\) gets close to \(0\text{.}\) What happens? Similarly, how does \(\frac{\sin(h)}{h}\) behave for small values of \(h\text{?}\) What does this tell us about \(AV_{[a,a+h]}\) for the sine function as \(h\) approaches \(0\text{?}\)
