Guided Practice 2.2: The sine and cosine functions

Overview

Now that we have general computational rules for power functions, exponential functions, constant multiples, and sums/differences of functions, we will add another family of functions to the toolbox: The trigonometric functions. In this section you will develop rules for the derivatives of the functions $y = \sin(x)$ and $y = \cos(x)$ as part of the activities in the class. Then we will use those rules in applied problems that involve angles and cyclic 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).

• (Trigonometry review) State the values of $\sin x$ and $\cos x$ at the angle values $0, \pi/6, \pi/4, \pi/3, \pi/2$ and other related points on the unit circle without a calculator. (Here is a good resource for trigonometry review, as well as a nice unit circle cheat sheet you can print off and carry around. There are also tons of great trigonometry resources on YouTube.)
• (Trigonometry review) State the definitions of the tangent, contangent, secant, and cosecant functions.

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.

• State the derivatives of the functions $y = \sin(x)$ and $y = \cos(x)$.
• Use the derivatives of $y = \sin(x)$ and $y = \cos(x)$ 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 2.2. Make sure to read actively, working through examples and activities as you go.

Video: Watch the following videos. These are just gathered from youtube.

• Unit Circle Exact Values: https://youtu.be/CJuLeDHSBEU
• Basic trigonometry: Sin, Cos, Tan https://www.youtube.com/watch?v=bSM7RNSbWhM
• How to remember the unit circle https://youtu.be/c819bGfH8FA

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.