​Learning Objectives​

• Connect the behavior of sine and cosine functions to their derivatives

• Evaluate derivatives that include sine and cosine

• Write equations of lines tangent to the sine and cosine graphs

​

Quick Lesson Plan

Experience First

In this lesson students will once again be using their toothpick tangents to study the slopes of sine and cosine. On the first page of the activity, students estimate the slopes of the sine function at various inputs. Students are prompted to consider where the slopes will be steepest and flattest and then use the provided values in the table to make conjectures about slopes at other points. Students will notice that slopes repeat cyclically, just like the original function! Plotting the slopes and connecting them with a smooth curve will allow them to see the cosine function.

 

On page 2, students almost immediately decide that the derivative of cosine has to be sine, but by looking at where the cosine is decreasing and increasing, students will notice that the slopes of cosine for x>0 are negative first. After plotting the slopes, students see that the derivative of cosine is actually opposite sine.
 

Formalize Later

When you debrief the front page, label the sketch of their slopes graph as y’. Have students describe the pattern of the slopes (starts at 1, then goes down to 0, then all the way to negative one, then gets flatter again back to 0, etc.) and then describe the pattern of the cosine function (starts at 1, then goes down to 0, then all the way to negative one, then back up to 0, etc.) This is a great opportunity to review key ideas of Lesson 9.6 about connecting the behavior of f and f’. When debriefing question 5, have students type in -sqrt(2)/2 in their calculator to realize this can be approximated as -0.707, which was the value in the table!

 

When debriefing the back page, demonstrate the fact that at an angle of pi/6, the slope of the tangent line to cosine is ½ and the output of the derivative equation is also ½, since the derivative is the slopes graph.

 

We use the language of “opposite” sine instead of negative sine as this presents a much clearer picture of what the “-” symbol means. We are trying to make this vocabulary switch in all of our math classes all the way down to Algebra 1.