Calculus I using LiveMath

Chapter 3

Derivative of the Sine and Cosine Functions

If you need the derivative of a trigonometric function in LiveMath, all you need do is input the function with the Derivative Op, select, and Simplify. If you want to understand the 'why's, you must look deeper. The difference quotient is the key, again, to finding the derivatives of these very important functions.

To derive these derivatives you must become familiar with the use of the trigonometric identities and use them in the derivations. By algebraically manipulating the difference quotient and using the identities found as Transformation Rules in LiveMath, you can do most of these derivations on your computer. To pursue this more rigorous study as it pertains to these functions, click here.

The fact that the derivative of the sine function is merely the cosine function, is one of several curious and fascinating tid-bits about calculus. This fact and, as you will see later on, the connection between the derivative and the integral, the derivative of exp(x) being equal to itself, and other interesting coincidences, may make you think that calculus is not the invention of man, but something innate to nature itself. It is as if all the answers to the questions of the universe are already in place, they are just waiting to be discovered and defined by man. It is the old argument of Platonism (where it is thought that math is discovered not invented) vs. Conceptualism (where natural occurrences are fit to mathematical statements because it is compelling).

The easiest way to see this fascinating relationship is to look at a Graph Theory with both functions plotted. Remember the "New" function called the Derivative Function gives the slopes, at values of x for the "Old" function. In the following graph, the Sine Function is plotted along with its derivative, the Cosine Function.

By using the same methodology used in Chapter 2, you will see how you can manipulate the Difference Quotient by changing the value of h. As h approaches zero, the derivative function approaches the Cosine Function and eventually coincides with it. The following example again shows how the power of the computer can make learning math both fun and easy.

3.12 Sine Derivatives or LiveMath Internet

Since you know that the derivative of the Sine is the Cosine, this example will examine just the Cosine and the difference quotient of the Sine function to show how they are one and the same, as h approaches zero.

• First input the Cosine function and graph it. Hit the Return key and input a new equation for the difference quotient of the Sine function using y' as the dependent variable. Hit the Return key again and input the equation h =.5 Now place the cursor inside the difference equation and choose Add Line Plot.
  
  When asked, define both y' and h as variables and define y' as the y-axis. Change the plot details so that a dotted line defines the difference equation rather than a dark line which is the default (or colored if you have a color monitor). If needed, use the following graphic as a guide for your inputs (the difference quotient plot is red).
  
• Change the value of h to watch the Difference Quotient approach the cosine function.

The cosine function, as you can imagine, must be similar, and it is. The derivative of the cosine function is merely the negative sine function.

From your study of trigonometry, remember that the difference between the plot of the sine function and the cosine function is just a shift along the x-axis of Pi/2. Wouldn't it follow that if you shifted the derivative function of the sine the same amount, you would have the derivative of the cosine. This is exactly the case.

• In LiveMath plot the derivative of sin(x) which equals the cos(x) function.
  
• Now add the derivative of cos(x) which equals - sin(x).
  
• Add a third function which is defined as the derivative of the sine function which is merely the cosine function. Add to it a translation parameter (a). To facilitate graph labeling it is best to set up the function as shown below.
  

Note: The sine function was left out of the Graph Theory just to keep the viewport uncluttered.

• Give a the value of 0 and Animate a from 0 to Pi/2. The resulting animation follows. As you can see, when the animation parameter is equal to 0 (shown in the upper right hand corner of the plot), the derivative of the sine function is equal to the cosine function. As you add to the parameter, the line gradually shifts over and turns into the derivative of the cosine function when a = Pi/2 (1.571).