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Handout Daily Prep 2.2 - The Sine and Cosine Functions

Section Overview

This section covers the following concepts: Gain an understanding of derivatives of the sine and cosine functions and their derivation.

Section Basic learning objectives

These are the tasks you should be able to perform with reasonable fluency when you arrive at our next class meeting. Important new vocabulary words are indicated in italics.
  • Identify the graph of a general exponential function \(f(x)=a^{x}\) and state its most important features (e.g. \(y\)-intercept, behavior for large positive or negative \(x\)-values). (See this website if review is needed.)
  • State the general rule for \(\displaystyle \frac{d}{dx}a^{x}\) (where \(a>0\) is real).
  • Identify what makes the function \(e^{x}\) special in terms of its relationship to its own derivative.
  • State the values of \(\sin x\) and \(\cos x\) at the angle values \(0\text{,}\) \(\pi/6\text{,}\) \(\pi/4\text{,}\) \(\pi/3\text{,}\) \(\pi/2\) and other related points on the unit circle without a calculator. (See this website for review if needed.)
  • State the derivatives of the functions \(y=\sin x\) and \(y= \cos x\text{.}\)

Section To prepare for class

Complete all actions listed below. Respond to the questions highlighted with Submit.
  • Read the motivating questions and the introduction to section 2.2 (up until Preview Activity 2.2.1).
  • 📝 [Submit] Do the following construction in GeoGebra. Submit screenshots and responses to the questions posed.
    • In the first input box, type sin(x). This will define and graph \(f(x)=\sin(x)\text{.}\)
    • To get a visual estimate of the derivative, we will use the definition of derivative which tells us that
      \begin{equation*} f'(x) = \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}. \end{equation*}
      Rather than take a limit, we will use a reasonably small value of \(h\text{,}\) say \(h = 0.01\text{.}\) Then,
      \begin{equation*} f'(x) \approx \frac{f(x+0.01)-f(x)}{0.01}. \end{equation*}
      In the next input box, type (f(x+0.01)-f(x))/0.01. This will define and graph
      \begin{equation*} \displaystyle g(x) =\frac{f(x+0.01)-f(x)}{0.01}\approx f'(x) = \frac{d}{dx}\sin x. \end{equation*}
      Have you seen the graph of \(g(x)\) before? Identify it and fill the following blank in accordingly:
      \begin{equation*} \frac{d}{dx}\sin x = \underline{\hspace{30mm}} \end{equation*}
    • Repeat this whole process to visualize the derivative (at least an estimate of it) of \(f(x) = \cos x\text{.}\)
  • 📝 [Submit] Do some exploration with GeoGebra to conjecture the value of the following limits:
    • \(\displaystyle \displaystyle \lim_{h \rightarrow 0}\frac{\cos h - 1}{h}\)
    • \(\displaystyle \displaystyle \lim_{h \rightarrow 0}\frac{\sin h}{h}\)
    One suggestion is to plot each of these expressions of \(h\) (namely \(\frac{\cos h - 1}{h}\) and \(\frac{\sin h}{h}\) - the behavior of each graph near \(h=0\) should indicate the value of each limit. [Important note: GeoGebra will not treat \(h\) as a variable. So, use \(x\) rather than \(h\) in order to explore each limit. That is, investigate \(\displaystyle \lim_{x \rightarrow 0}\frac{\cos x - 1}{x}\) rather than \(\displaystyle \lim_{h \rightarrow 0}\frac{\cos h - 1}{h}\text{.}\)] Submission of screenshots would be ideal.
  • Prompt Copilot “Why is the derivative of \(\sin(x)\) only \(\cos(x)\) when \(x\) is measured in radians and not degrees?”
  • First, try to answer this question before prompting the AI. Then, prompt Copilot “Can you give me an example of a function whose derivative is periodic but it itself is not periodic?” Is the response reasonable?

Checkpoint 56. 📝 [Submit] Ideas Behind the Derivatives of Sine and Cosine.

Section After class

Solidifying the concepts discussed in class through practice is necessary to build your skills.

Section Advanced learning objectives

In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
  • Use the derivatives of \(y=\sin(x)\) and \(y=\cos(x)\) in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.
  • Prove that \(\displaystyle \frac{d}{dx}\sin x=\cos x\text{.}\)

Section Additional suggestions

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