12/2001

Trigonometric Functions

Purpose: The purpose of this lab is to discover the derivatives of the basic trig functions.

Experiment: Recall what the graphs of the sine and cosine functions look like:

Note the values of each function for t = 0; p/2 ; p .

Now we will try to discover what the derivatives of these functions are. For each function, we will ask Mathcad to find its derivative and graph it, and see if we recognize the resulting function:

Look carefully at these results and make a conjecture:

The derivative of sin(t) is...

The derivative of cos(t) is...

Check your conjecture by graphing your new function along with the derivative in each graph above. When the graphs coincide, you have found the correct answer.

Now we will look at the families of functions

f(t) = a cos(bt) g(t) = a sin(bt)

where a and b range over the real numbers. Initially a and b are both 1, so we have the same functions as our original ones. Here are the graphs of f and g and their derivatives:

Varying the amplitude: Experiment with varying the constant a. Choose both positive and negative numbers. How does this affect f, g, and their derivatives?

Varying the period: Now experiment with varying the constant b. Choose both numbers less than 1 and greater than 1. How does this affect f, g, and their derivatives? (You may need to adjust the scale on the vertical axes.) Use the Chain Rule to explain your answer.

On a sheet of paper, write down these functions and find their derivatives. (Or print out this page, and write on it.) Turn in the sheet with your lab (one per team):

3 cos t

4 sin t

cos 5t

2 sin 6t

cos t sin t

cos²(6t)

e^t cost

ln t sin t

Now switch typists.

Other trigonometric functions:

Two other basic trig functions are the tangent and secant functions:

Explain what those vertical red lines in the graphs are. Why are they there? Where exactly do they occur?

Here are the graphs of the derivatives of the tangent and secant function:

We will use our knowledge of trigonometry and differentiation to discover

what those derivatives are.

On a sheet of paper, write tan(t) as a ratio of two other trig functions.

Use the Quotient Rule to find its derivative. Check your answer by

graphing it on top of the graph of Dtan(t) above; if they agree, you have

found the derivative correctly. (By the way, Mathcad does not understand our shorthand notation sin²t or cos²t; you have to write sin(t)², for instance.)

Can you simplify your answer?

Do the same thing for the derivative of sec(t). When the graph of your

function matches the graph of Dsec(t), you have the correct answer.

As a group, write a maximum of one page discussing the topics covered in this lab. Explain what you have learned and how it fits in with the bigger picture of the calculus you've studied so far in this class. Hand in the writeup with your lab.