Midterm 1 Preparation: Math Topics and Examples for Sin, Cos, Tan Functions - Prof. C. Pha, Study notes of Mathematics

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Homework:

• Section 6.5A: Handout
• Section 6.6 (pp. 479): 4, 6, 8, 10, 14, 18, 26, 30, 34, 40, 44, 48, 52, 54. Due: Wednesday, 1 February 2006. (For your own study purposes, you should complete problem 20, but you needn’t turn it in.)

Midterm #1 information

Yes, time flies when you are having fun, and it is already time to consider our first midterm. The examination will take place Friday, 3 February 2006 , for the entire class period. We will dedicate Wednesday to review. The test will cover material up through §6.6. There will be two types of questions: short answer and open response.

• Here is an example of a short answer question: “What is sin π/ 4 ?” On these questions, it is not necessary to show your work; however, I will expect exact answers only. Calculators will not be allowed (or necessary) on this section of the exam.^1
• Here is an example of an open response question: “Show that (^) cos^1 t − sin t tan t = cos t.” Here are some topics with which you should be familiar:

1. Angles : You should understand what standard position is. You should understand what it means for angles to be coterminal.
2. Radians : You should be familiar with radian measure. You need to be able to draw angles of the following radian measures in standard position: 0 , π/ 6 , π/ 4 , π/ 3 , π/ 2 , 2 π/ 3 , 3 π/ 4 , 5 π/ 6 , π, 7 π/ 6 , 5 π/ 4 , 4 π/ 3 , 3 π/ 2 , 5 π/ 3 , 7 π/ 8 , 11 π/ 6. You should also be able to identify angles which are coterminal to these (such as an angle with radian measure 44 π/ 6 ) and draw these as well. ( Hint : You shouldn’t memorize all these angles. You really only need to know how to draw angles with radian measure 0 , π/ 6 , π/ 4 , π/ 3 and π/ 2. The rest can be obtained by reflecting over either the x- or y-axis.) You should also understand the definition of radian measure (provided in bold on p. 414).
3. Arc length : You should understand the relationship between the radian measure of an angle and the length of an arc it subtends on a circle with arbitrary radius.
4. Linear speed and angular speed : You should understand the relationship between these.
5. The sine, cosine, and tangent functions : You should be familiar with the definitions and graphs of our three new friends, the functions sin, cos, and tan. You should be able to evaluate these functions at all the radian measures listed in item 2 above. You should understand the periodicity properties of these functions, and be able to use these to evaluate these functions at values such as 93 π/ 4. You should understand and be able to use the point-in-the-plane description of these functions.
6. Identities : You should be familiar with our two (admittedly annoying) conventions, and not be confused by them. You should understand what an identity is. You should be familiar with the pythagorean identity, the periodicity identities, and the negative angle identities. From Math 111 (or an equivalent course in your past), you should know what it means for a function to be odd and even ; from this course, you should be able to related these concepts to the negative angle identities, or use the negative angle identities to prove that a function (such as f (t) = t + tan t) is odd, even, or neither. (If you don’t remember odd and even from Math 111, don’t feel bad; just make sure you understand them now. These are discussed in §3.4A. (^1) The problem with allowing calculators on this portion of the exam is that it makes these problems trivially easy to anyone who is using a TI-89 or equivalent calculator.

7. Graphs : You should be able to graph the functions sin, cos, tan, sec, csc, cot without a calcula- tor. You should understand how these graphs are obtained. You should also be able to apply transformations (§3.4). You should understand the connection between graphs and identities
8. Periodic graphs and simple harmonic motion. You should be able to explain the period, amplitude, and phase shift of functions of the form f (t) = A sin(bt+c) and g(t) = A cos(bt+c). You should be able to apply these to applications involving spring-mass systems, pendulums, and so on.
9. Dampened harmonic motion. You should be able to sketch a graph of functions of the form f (t) = Ae−dt^ cos(bt). You should be able to answer questions about motion that follows such a function.
10. The cotangent, secant, and cosecant functions. You should be familiar with the definition, graphs, periods, and point-in-the-plane descriptions of the functions cot, sec, and csc. You should also be familiar with the reciprocal and pythagorean identities involving these func- tions. You need to be able to evaluate these functions at the values listed in item 2, as well as be able to use the periodicity properties to evaluate them at other values (such as − 91 π/ 3 ).