At the end of this session you will be able to
In the lessons up to know, we have learned what a derivative is, learned how to differentiate using rules for sum, difference, product, quotient, and power functions, and learned applications of the derivative. In this lesson we will learn how to differentiate trigonometric function and, in the process, review the differentiation you learned previously.
DERIVATIVE OF THE SINE FUNCTION
The trigonometric identities found on pages 53 and 54 are the foundations for the learning in this unit. If you do not know them, then take the time now to learn them.
As you study the material at the top of page 180 of Section 3.4, you see that the derivation requires Theorem 7 and Example 5(a) from pages 105 and 106 and the addition identity from page 53. Now that you have studied the material from Section 3.4 of your textbook, you know the derivative of
DERIVATIVES INVOLVING THE SINE FUNCTION
When you practiced Example 1 on page 184, you were putting the knowledge about the derivative of a sine together with the other differentiation rules you learned. We are building on what we learned before!
Now you can go to the Session 9 assignments to continue your assignment.
At the end of this session you will be able to
DERIVATIVE OF THE COSINE FUNCTION
The trigonometric identities found pages 53 and 54 are the foundations for the learning in this unit. If you do not know them, then take the time now to learn them.
As you study the material at the top of page 185 of Section 3.4, you see that this derivation also requires Theorem 7 and Example 5(a) from pages 105 and 106 and the addition identity from page 53. Now that you have studied the material from Section 3.4 of your textbook, you know the derivative of the cosine is
Now looking at the rules for the derivative of the sine and the derivative of the cosine, you should see the potential of missing the minus sign on the derivative of the cosine. So, be careful.
DERIVATIVES INVOLVING THE COSINE FUNCTION
When you practiced Example 2 on page 185, you were putting the knowledge about the derivative of a sine together with the other differentiation rules you learned, including the derivative of the sine. We are building even more on what we learned before!
Now you can go to the Session 9 assignments to continue your assignment.
At the end of this session you will be able to
CALCULATE VELOCITY AND ACCELERATION
If you have forgotten what amplitude and period are for trigonometric functions, then you will find help on pages 52 through 53.
If you did not study slowly and meticulously page 186 of Section 3.4, then returning to study these pages again will help immensely Your textbook authors packed considerable information into those pages.
Now you can go to the Session 9 assignments to continue your assignment.
At the end of this session you will be able to
DERIVATIVE OF THE OTHER TRIGONOMETRIC FUNCTIONS
The derivation of the derivative of all the other four trigonometric functions use the derivative of the quotient and the definition of these four trigonometric functions. The definitions of the tangent, cotangent, secant, and cosecant functions are given on page 187. Notice that all are expressed as quotients.
Therefore, the derivatives of the four trigonometric functions (3), (4), (5) and (6) all result from using the quotient rule as illustrated in Example 5 on page 187 and page 188. On your own try differentiation the secant, cotangent, or cosecant using the quotient rule to obtain the derivative as your textbook did in Example 5.
DERIVATIVES INVOLVING THE OTHER TRIGONOMETRIC FUNCTIONS
In your textbook on page 187, Example 6 illustrates putting all of these differentiation formulas into action. The assigned practice problems will do the same.
Before, we learned that one of the applications of differentiability was to test for continuity. The top of page 188 shows that differentiability of trigonometric functions also guarantees continuity for trigonometric functions within the domains of the trigonometric functions.
Now you can go to the Session 9 assignments to continue your assignment.