Education Technology

Discovering the Derivative of the Sine and Cosine Functions

Published on 06/09/2008

Activity Overview

In this activity, students discover the derivative of sin(x) and cos(x) by analyzing a scatter plot of x-values and the function's numerical derivatives at these x-values.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 123 - 126 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:

  • Define f(x) = sin(x), x1 = 1, and y1(x) = f(x)
  • Store x1 in L1 and derivative of f(x1) in L2
  • Set up a scatter plot
  • Sketch the graph on grid provided
  • Change values of x from 1 to {x, x, -2π, 2π, π/12} , {0, 1, 2, 3, 4, 5, 6} and { -6, -5 , -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 }, and add new slopes to the scatter plot on the grid
  • Define a function y2(x) that will connect the data in the scatter plot
  • Observe that the function that connects the data in the scatter plot is cos(x)
  • Define y3(x) as the derivative of f(x), use nDeriv function to find the derivative of f(x)
  • Note that y2(x) = y3(x)
  • Observe that if f(x) = sin(x) then f'(x) = cos(x)
  • Redefine f(x) = cos(x) and observe the effect on the graph
  • Observe that the graph of f(x) and the scatter plot of the numerical derivatives have changed
  • Also note, since y2 = cos(x), the graph of f(x) is the same as the graph of y2
  • Again define a function y2(x) that will connect the data in the scatter plot
  • Observe that the function that connects the data in the scatter plot is -sin(x)
  • Observe that if f(x) = cos(x) then f'(x) = -sin(x)
  • After the Activity

    Students complete the Student Activity sheet.

    Review student results:

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary