Guided Practice 2.4: Derivatives of other trigonometric functions

Overview

In this section, we will use the Product and Quotient Rules along with our rules for differentiating $\sin(x)$ and $\cos(x)$ to derive formulas for the derivatives of the remaining four trigonometric functions: $\tan(x)$, $\cot(x)$, $\sec(x)$, and $\csc(x)$. We will then use these trigonometric derivatives in problems involving angles and cyclical behavior.

Learning objectives

Basic objectives: Each student is responsible for gaining proficiency with each of these tasks prior to engaging in class discussions, through the use of the learning resources (below) and through the working of exercises (also below).

• (Review) State the definitions of the tangent, secant, cosecant, and cotangent functions.
• (Review) State where the sine and cosine functions attain a value of 0.
• (Review) State the “Fundamental Trigonometric Identity”. (Hint: Look on page 113.)

Advanced objectives: The following objectives are the subject of class discussion and further work; they should be mastered by each student during and following class discussions.

• Derive the rules for differentiating $\tan(x)$, $\cot(x)$, $\sec(x)$, and $\csc(x)$ “from scratch” using only basic derivative rules and trigonometric identities.
• State the derivatives of $\tan(x)$, $\cot(x)$, $\sec(x)$, and $\csc(x)$.
• Use the above rules in the context of a real-world problem to find the slope of a tangent line, the instantaneous rate of change in a function, or the instantaneous velocity of an object.

Learning resources

To gain proficiency in the learning objectives, use the following resources. You may include other resources if you wish, in addition to or in replacement of the following.

Textbook: In Active Calculus, read Section 2.4. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos at the MTH 201 YouTube playlist (http://bit.ly/GVSUCalculus). Note: Before viewing these, you may first need to refresh yourself on the definitions of the trigonometric functions that are involved. It’s up to you how you want to do this – through reading on the web, watching other videos found through a YouTube search, or just wing it. But be honest with yourself – if you are not understanding the trig, stop and find help.

• Quick Review: Derivatives of other trigonometric functions (1:56)
• Examples of other trig derivatives (9:47)

Activities

The following activity is to be done during and following your reading and viewing of the resources. Go to student.desmos.com and join using your name in the format Last, First. For example, I would enter my name as Ballif, Serge. Complete each part of the activity. Some of these problems will require you to work them out on paper before entering your answer. Practice producing high quality work so that your work is readable and meaningful. You will receive a mark of Pass if each item response shows a good-faith effort to be right and is submitted prior to the deadline. Remember to use the Piazza discussion board to ask about any questions you have.