Last Updated: 8/21/17

Math 95 Page Contents

• Course Information Sheet
• Student Learning Outcomes for Math 95
• Textbook Info
• Homework Assignments 
• Links of Interest

Course Information Sheet

Here are links to the Course Information Sheets for my Fall 2017 sections of Math 95: 10-11am (section 3) and 2-3pm (section 7).

Please read the Course Information Sheet carefully...it provides all the important information about how the course works (textbook info, office hours, details of policies regarding exams, quizzes, homework, grading, etc.). It will be here for you to refer to throughout the semester.

Student Learning Outcomes for Math 95

Textbook Info

• Trigonometry: A Unit Circle Approach, by Michael Sullivan (Custom Edition for City College of San Francisco).

Note that, in our section of Math 95, you are not required to purchase the "MyMathLab" support materials...we wil not be using their online homework software. But if you end up getting the MyMathLab package anyway, and you want to access its support materials, let me know and I'll assist you if need be.

You can purchase our course textbook in any one of the following ways:

• (Recommended) Going to the college bookstore to buy the text. Note that the bookstore offers a special CCSF Edition, culled from the 9th edition, and including only the topics in our CCSF Math 95 course outline.
• (On your own) Buying the book elsewhere (e.g. online). If you go this route, make sure you get the 9th edition and the "Unit Circle Approach" version of our text.

Students should obtain their textbook and have it available for study starting on the first day of classes. If you don't have a textbook by the first day of class, there are also a number of copies of the textbook on 2-hour (or more) reserve in the college library (e.g. if you need to read the book while waiting for delivery of a text purchased online).

If buying textbooks is a significant financial burden, look into the free CCSF Student Bookloan Program.

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Students are highly encouraged to look over the material in Chapter 1 as well as the several appendices A1-A8 at the end of the book. Looking through this material may prompt you to review various topics with which you may be "rusty"... or at the very least make you are aware of where to locate review info if you need to revisit it later in the course. Though this material may not be fresh in your mind, most of it should be familiar, and you should be able to review it and get up to speed fairly quickly for this course. If that's not the case, you should likely be enrolling in Math 90 (Precalculus Algebra) rather than Math 95.

Homework Assignments

Below is the list of HW assignments and associated commentary, so far:

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Trigonometry is the study of the six "trigonometric functions" and their many properties and uses. Thus understanding trigonmetry requires familiarity with the concept of a function and many other ideas associated with the function concept (e.g. domain, range, maxima, minima, etc.), which you should be familiar with from your prerequisite study of functions in algebra. The fundamentals we'll need are reviewed in §1.3 – §1.7 of our text...you should review those sections as needed.

It turns out that, in a certain natural sense, the inputs for the six trigonometric functions are angles (or more precisely, the numbers representing the measure of geometric angles).

So we'll begin by reviewing various aspects of the concept of an angle. You should be familiar with measuring angles in terms of "degrees" (i.e. 30°, 90°, etc.). Becoming familiar with measuring angles in terms of another system of units, called radians, is a critical skill we'll need to master. Read §2.1 of the text and write up solutions to the following exercises from our textbook:

• §2.1: #1-10 all, 11-23 odd, 29, 35-59 odd, 67, 71-77 odd, 87, 89, 99, 101, 103, 107, 114.
• The following definition will be very useful for us: a "special angle" is any integer multiple of either 30° (= π/6 rad) or 45° (= π/4 rad). The special angles between 0° to 360° appear in degrees in this diagram and in radians in this diagram (see also the bottom of pg. 173 of our text). A blank circle diagram appears here...you can use it to practice labelling the terminal rays for the "special angles". Eventually, you'll need to memorize these.

Due Monday, August 28: Solutions to the exercises assigned above from §2.1 .

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• "The Right Triangle Approach": Before proceeding to §2.2 of our textbook, we will briefly discuss similar triangles, and the Pythagorean Theorem, and most importantly... the "Right-Triangle Approach" to defining the six trigonometric functions. The main ideas we'll focus on are summarized in the class handouts below:

  • (Handout #1) "Right Triangle Trig: Basic Definitions (and Some Drill Exercises)" (PDF here [120KB]);
  • (Handout #2)"Basic Right Triangle Exercises" Classroom Handout (PDF here [625KB]);

Due Wednesday, September 6: (Note: Monday is a holiday...no classes!) Turn in solutions to exercises on Handouts #1 and #2 (i.e. the two PDF's above). Write your solutions on separate sheets of paper, as there's not enough blank space on the handout sheets.

• "The Right Triangle Approach (continued)": Will explore some additional ideas relating to the "right triangle approach" in the class handouts below:
  • (Handout #3) "Using Right Triangles to Tabulate Sine and Cosine Values" (PDF here [68 KB]).
  • (Handout #4) "The Three Unit Right Triangles".

  For some alternative discussions of the "Right-Triangle Approach" to trigonometry:

  • Visit Dr. Carol JVF Burns' web page here ;
  • Study pages 13-18 of the well-written document Trigonometric Functions and Triangles, by Phillipe B. Laval (available as a PDF file at this link).
  • Visit David Joyce's web page on the topic.

  Note that the first 10 pages of Laval's paper overlap with material from §2.1 of our text. In particular, Laval introduces the helpful concept of a "reference angle" (on pages 6-7)...which though not discussed in our textbook, is a very useful notion.

Due Monday, September 11: Turn in solutions to exercises from Handouts #3 and #4 (PDF's above). Write your solutions on separate sheets of paper, as there's not enough blank space on the handout sheets.

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The next section, §2.2, is fundamental to the course. It introduces "the unit circle approach" to Trigonometry. We will discuss this in detail in class, and you should carefully read (and reread as necessary!) this section and memorize the defintions of the six trig function outputs of a given input angle, as defined in the "unit circle" context.

• § 2.2: #1, 9, 10, 11-31 odd, 39, 41, 49, 53, 55, 59, 63, 83, 85, 97, 99, 128. [Note: Look here for a "Unit Circle Diagram" link which may be helpful for this HW assignment...and in the future. ]

• § 2.3: #1-11 all, 17-37 odd, 43, 45, 49, 51, 53, 55, 59, 69, 93, 113.

• Trig Functions by Hand: You can download the class handout entitled "Trig Functions by Hand" here (PDF 265 KB).

  At our next class meeting on Monday, 9/25 we will discuss how to use the "Unit Circle + Tangent/Cotangent Axis Diagrams" on that handout to visually evaluate the six trig functions by measuring various lengths.

  In preparation for that, read the items listed below, and study the handouts appearing there as PDFs.

  • Note 1: Do not use a calculator for exercises a)-d) on the handout! The point is to use the relationships illustrated in the "diagrams" at the bottom left of the handout to estimate the trig function values by measuring various lengths (to, say, 2 places beyond the decimal point...your answer will not be exact).
  • Note 2: The ideas explored on the handout are illustrated in this interactive "Geogebra applet" (it may take a few moments to load). See also, this similar Geogebra project created by CCSF Math 95 student Chris Hartford. Once the applets are loaded in your web browser window, carefully click on the red point labelled B and drag it along the unit circle. Note how all the colored lengths change (i.e. sinθ, cosθ, tanθ, secθ, cotθ, and cotθ) as the input θ changes when you move the red point B. That red dot corresponds to the point on the unit circle that we've denoted by P_θ in class. The ideas here are also discussed on this webpage (which also has a similar interactive applet).
  • Note 3: I've written a detailed explanation of the ideas behind the "Trig Functions by Hand" worksheet (and the interactive applets) above. I titled the handout "Visualizing the Outputs of the Six Trig Functions", and it can be downloaded here (PDF 1.4 MB)

    Also, if you want to practice using the "Unit Circle + Tangent Axis + Cotangent Diagram" to evaluate the 6 trig functions for other θ-values, you can download and print out a "blank copy" here (PDF 147 KB).

  [For the answers to the "Trig Functions by Hand" exercises click this link...but the answers themselves aren't the "goal" here, the goal is to understand how to use the "tangent axis" diagram to obtain the values of tan(θ) and sec(θ) by measuring the appropriate lengths; and similarly to understand how to use the "cotangent axis" diagram to get values for cot(θ) and csc(θ) ].

• § 2.4: #3-11 all, 17, 21, 23, 29, 35, 39, 51, 63, 65, 67, 51, 71.

Note that besides learning the basic facts about the graphs of y = sin(x) and y = cos(x) ...like period, amplitude, basic shapes...much of the material in §2.4 is really about vertical and horizontal "scalings" of graphs (see §1.6 of our text for a review of those topics).

In §2.5; and §2.6 we'll also encounter vertical and horizontal "shifts" of trig graphs. Our end goal will be to understand how to graph and interpret any function of the form:   f(x) = A trig(b x + c) + d .  Here A, b, c, and d are constants and "trig" stands for any one of the six trigonometric functions.

All these "scalings and shifts" of trig graphs are just a special case of what are usually referred to as "Geometric Transformations" for graphs of (arbitrary) real-valued functions of a real number input variable. This topic is reviewed in §1.6.

• § 2.5: #3-6 all, 11-17 odd, 18, 21, 23, 27, 29.
Here are two nice supplemental items you should study:
• A Desmos module illustrating how the graphs of the six basic trig function arise from the "Unit Circle Approach"
• A "Geogebra" applet illustrating the "Tangent and Cotangent Axis" approach for visualizing the six trig function values (created by a student in our trig class! :-).
• § 2.6: #1, 3, 5, 11, 15-23 odd, 33, 35.

• § 3.1: #7-12 all, 13, 15, 16, 19, 21, 23, 29, 37, 39, 41, 42, 45, 47, 49, 53, 76.

• § 3.2: #1, 2, 3, 9, 13, 17, 21, 25, 27, 29, 33, 37, 45, 47, 57, 59, 61, 65. (NOTE: Problems #37, 45, 47, 61, and 65 are optional: they involve the inverses of sec or cot, which will not be discussed in detail in class. They're provided for students who may be curious and interested in trying problems involving inverses of sec, cot, etc.)

Due Monday, November 6: Finish writing up solutions to exercises assigned from §3.2.
Also, prepare for our exam next Wednesday, using the "Topics List for Exam 2" handout from class.
In addition to reviewing the material from §2.4-2.6 (on rectangular coordinate graphs of trig functions), practice being able to do problems like #13-52 from §3.1; and problems #4-66 from §3.2.

After Wednesday's exam, we will continue on in Chapter 3...the upcoming HW appears below:

• § 3.3: #7-11 all, 13, 14, 18, 21, 25, 33, 35, 37, 41, 45, 49, 53, 57, 59, 63, 66, 77.
• § 3.4: #1-9 all, 11, 13, 15, 23, 27, 29, 33, 37, 39, 49, 53, 75.

• § 3.5: #5-11 all, 13, 17, 25, 27, 33, 35, 39, 45, 51, 53, 61, 75, 87.
• § 3.6: #1, 4, 5, 7, 47, 69, 83, 84.
• § 4.1: #1-5 all, 9, 29, 37, 41, 49, 51.

Note: We'll discuss §4.3 (Law of Cosines) first, and after that §4.2 (Law of Sines).

• § 4.3 (Law of Cosines): #3-8 all, 9, 11, 17, 29, 33, 37, 42.
• § 4.2 (Law of Sines): #3, 4, 5, 7, 8, 9, 14, 21, 25, 35, 38, 47.

Note: A list of review topics for our Math 95 Final Exam appears here.

Links of Interest for Math 95

• Here's a link to a nice Unit Circle Diagram showing the terminal rays for all the "elementary angles" together with the coordinates of the corresponding points on the unit circle. You should become very familiar with the material in this diagram. (BTW, I really like their "unit circle T-shirt" design...but note that those will not be allowed during our Trig class exams!! :-)
• This "Unit-circle" Trigonometry PDF document provides an good development and presentation of the basic concepts of that topic. (But note that it works with angles in degrees, whereas we'll work mainly with angles in radians.)
• Dave's Short Trig Course is a very nice supplementary resource for your study of trigonometry. The diagram at the top of the opening page shows the natural relationships between the sine, cosine, tangent, and cotangent functions and basic triangles associated with the unit circle. Note also that nearly all the figures on the site are "dynamic", meaning you can "click and drag" on certain points and see the diagrams change into different configurations (while still maintaining the "relationships" under consideration). "Dave's Short Trig Course" differs from our course text in that it begins with a "triangle approach" (rather than the unit circle approach) ...so reading and studying his development of the basic concepts of trig can provide an alternate viewpoint to our book (and experimenting with the many interactive graphics can be enjoyable & enlightening).
• Desmos is a very "dynamic" web-broswer based graphing utility. It provides lots of opportunity to explore various graphical aspects of trig. I've used it in class to illustrate various concepts, and I encourage you to experiment with it on your own as well. It may help you develop and expand your understanding...and have some fun playing with ideas.
• People have discovered many proofs of the Pythagorean Theorem. How many? This question was posed to "Dr. Math" at the Math Forum website. Dr. Math refers to the Pythagorean Theorem webpage on Alex Bogomolny's "Cut-the-Knot" math website, which gives details for many proofs of the Pythagorean Theorem. Some of the proofs on that webpage are illustrated with fun interactive Java applets...where you can click and drag various points of a diagram and modify its configuration...bringing the ideas behind the constructions to life. You'll have to scroll down the page past Alex's preliminary remarks to get to the proofs.

More links for Math 95 can be found within the "Math Links Page". Here is a shortcut to the trigonometry links on that page.