Math 141-142: Unit 4

Topics

In this unit, you will study the graphs of tangent, cotangent, secant, and cosecant, with variations. You will also study the inverse trigonometric functions. The textbook is quite brief in these three sections, so I've include some supplementary material and exercises, as indicated below in the Study Guidelines.

• Graphs of variations of tangent, cotangent, secant, and cosecant (5.5)
  • Period and phase shift
• Inverse trigonometric functions: arcsin, arccos, arctan (6.1-2)
  • Computations
  • Graphs

Study guidelines

These reading and problem assignments are designed to help you learn the course material. You should complete all of these problems, check your answers in the back of the textbook, and get help with the problems that you missed. Most of the problems are odd-numbered, so you can check the solutions in the Solutions Manual.

The only way to learn mathematics is to do mathematics, so while these problems will not be collected or graded, you will probably not do well in the course if you do not complete these and check your work as described above. After completing these problems, go on to the Unit Exam Description below and follow directions.

• Pages 164-165 (review): Asymptotes
  • Reading: pages 164-165 on vertical and horizontal asymptotes
    
    
• Section 5.5: Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
  • Reading: section 5.5
    Read and work through examples 1-3 and their matched problems.
  • The textbook neglects any mention of period and phase shift for these four functions. But, just as in section 5.6 for sin and cos, you should be able to determine period and phase shift for variations of these functions. See the supplementary material on period and phase shift for definitions and exercises.
  • You can also try out a java applet that illustrates period, phase shift, and vertical stretching of the graphs of tangent, secant, cotangent, and cosecant.
  • You may of course use your graphing calculator to help graph these functions, but it is also a good idea to be able to do at least a rough sketch by hand. Be sure to take period and phase shift into account when graphing or recognizing graphs of these functions.
  • Practice Problems: 5.5 #1-35 odd
  • Additional exercises on period and phase shift
    
    
• Section 4.1 (review): Inverse functions
  • Reading: section 4.1
  • For a quick review, see the module on inverse functions. This module includes discussion of the concept, examples, and several animations and applets.
  • Problems: Work through a representative sampling of the problems in this section until you feel comfortable with the material.
    
    
• Section 6.1: The Inverse Sine, Cosine, and Tangent Functions
  • Reading: section 6.1
    Read and work through examples 1-8 and their matched problems.
  • This particular book unfortunately uses the sin^-1x notation for the inverse trig functions. As noted on page 389, this notation can cause confusion because the -1 exponent is not really an exponent, it's just notation. Therefore, I encourage you to use the more standard notations: arcsin x, arccos x, and arctan x, and I have also used these in the exams. You really should be familiar with both notations.
  • Pay close attention to the definition of the inverse trig functions, particularly the range of these functions:
    • The range of arcsin x is the interval [-pi/2,pi/2].
    • The range of arccos x is the interval [0,pi].
    • The range of arctan x is the interval (-pi/2,pi/2).
  • You can also try out a java applet to further explore the definitions of the inverse trig functions.
  • The textbook does not ask any questions involving graphs of the inverse trig functions. Thus, I've installed a practice assignment (on the MapleTA testing system) called Practice: Graphs of Inverse Trig Functions to give you some exercises in this area.
  • Practice Problems: 6.1 #1-45 odd, 51
  • Additional practice exercises on graphs of inverse trig functions:
    • At the testing web site for Math 141 or Math 142, choose Practice: Graphs of Inverse Trig Functions. You may take this as many times as you like - you will see different questions each time.
      
      
• Section 6.2: The Inverse Trigonometric Functions (continued)
  • Reading: section 6.2
    Read and work through examples 1-3 and their matched problems.
  • The arccot and arccsc functions are almost never used. However, arcsec is useful in calculus. Although I have assigned a few exercises involving arcsec, you will not be asked any questions on the exam about these three functions.
  • Practice Problems: 6.2 #1-27 odd, 33, 37, 50
    
    
• Supplementary material (optional) for sections 5.5 and 6.1-2:
  • Student Solutions Manual
  • For tutoring help, visit the Prentice Hall Math Tutor Center. Tutors can be contacted by phone, fax, or e-mail. To register, you will need to first obtain an access code from me.
  • Graphing Calculator Help

Unit Exam #4

After completing the above work, do the following:

1. Read the exam description:
   • This exam has 20 questions, and will count 20 points toward your grade. 
   • This exam has a one hour time limit.
   • 12 of the 20 questions are on section 5.9.
   • Many of the questions on this exam are multiple choice. For other questions, be sure to look under the entry box for the type of answer expected.
     • Some questions expect several answers. You have to enter these in the order requested in the problem, separated by commas. For example, if the question is What are cos x, sin x, tan x?, your answer might be 1/2,sqrt(3)/2,sqrt(3).
     • Other questions on this exam ask for an approximation instead of an exact answer. These questions will ask you to give an answer "with an accuracy of at least two decimals after the decimal point" (or possibly one decimal, or three decimals, etc.). In this case, you would have to enter at least two digits after the decimal point. For example, if you calculate your answer to be 1.41421356237, then you could enter 1.41, 1.414, 1.4142, etc. When you view your corrected exam, you will see that the "correct" answer is 1.41421356237  0.01. The symbol means "plus or minus", so anything within 0.01 of the first number counts as correct.
       • If a question does not ask for an approximation, then your answer must be exact.
   • Make sure your calculator is set to the correct mode (radians or degrees) for each problem. All inverse trig functions should be calculated in radians unless specifically stated otherwise.
   • This exam will be much easier if you are good at using the graphing features of your calculator. However, it will slow you down if you depend too much on your calculator. Be sure to consider basic properties such as periods and phase shifts when you are dealing with questions about graphs.
   • In previous classes, the more difficult topics were: (1) problems involving phase shifts of tan, cot, sec, and csc, and (2) problems like #1-28 in section 6.2.
2. Complete the online Unit 4 Pretest assignment for Math 141 or Math 142. You may use your book if you wish, and redo the pretest as many times as you like. Your pretest score will be scaled to 5 points maximum.
   • Directions: Click on the link above for your class, then choose the Unit 4 Pretest.
   • The pretest must be completed by the deadline date listed at the top of this page.
     However, you may redo the pretest as many times as you like before the due date.
     Your best score counts, and it will be rescaled to 5 points maximum.
3. If you are having trouble with any of the problems listed above or on the pretest or practice exams, make use of the help resources listed on the Help page.
4. Arrange with your proctor to take the online proctored Unit 4 Exam assignment for Math 141 or Math 142. Remember to bring identification, and remember that you will not be able to take the unit exam after the deadline date given at the top of this page. You may NOT use your book or notes on this exam.
   • Directions: Click on the link above for your class, then choose Unit 4 Exam.
   • The proctored unit exam must be completed by the deadline date listed at the top of this page, and may be repeated under certain conditions. See the Detailed Schedule page for Math 141 or Math 142 for specific rules.

   Note: Remember that you can always go back and take practice exams on this unit for Math 141 or Math 142 after the deadline has passed. In particular, this will help you prepare for the midterm and final exams.

   • Directions: Click on the link above for your class, then choose Unit 4 Exam. After the deadline has passed, this exam will be available in practice mode.

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