Each class will begin with whatever announcements need to be made, and then I'll cut you loose to work on your homework and test preps.

If you want me to cover a particular concept/example/exercise during the face-to-face, hit me up in E-Mail (Use Classlist on D2L), and I can prep specifically for your question or even make an extra video ahead of time, in some cases.

• 1/20 Notes
  • LOL! I actually called a meeting on MLK, Jr. holiday! DOH! There's some good stuff in there, but probably nothing that isn't covered in Wednesday, 1/22 lecture. *sigh* Things were pretty laid-back, and I think the 4 over-achievers who showed up got something out of it (starting with my apology).

    I'm sorry that I didn't record the lecture. I try to reduce the length of recordings by pausing the recording and then re-starting it, as needed. I just messed up the first day. :o(

• 1/22 Notes
• 1/22 Video
  • We had a half-decent informal orientation for everyone on the actual first day of class!

    We defined radian measure and showed its relationship to arc length. We got a rough idea of the quadrant angles as decimal radians, relating them to better-known degree measure:
    1.57 <-> 90 degrees, 3.14 <-> 180 degrees, 4.71 <-> 270 degrees, and 0 <-> 6.28 <-> 0 degrees or 360 degrees.

• 1/27 Notes
• 1/27 Video
  • We covered the 5 triangles that should carry you through the whole semester, instead of trying to memorize the 12-point unit circle. Good stuff. Ask me about it, next time.
• 1/29 Notes
• 1/29 Video
  • Worked a version of WA#2 #1, using period to help evaluate a trig function of a large angle (bigger than 2Pi).

    Worked a version of the "height of the mountain" question.

    Found the values of the 6 trig functions of a point on the unit circle. Didn't mean to, but worked the same version as WA#2's. This is one I had to explain twice, because I was sharing the wrong screen! DOH! Thanks for the heads-up Alexis!

    Discussed even/odd, briefly.

• 2/3 Notes
• 2/3 Video
  • Week 2 Late Edition open until Friday. I will open up another version of it after I upload solutions. The 2nd version will be for 30% deduction. The first version (of Late Edition) will be no deduction until I upload.

    Hope that makes sense.

    To "see" an angle in radians, I always convert to degrees via (180 degrees/Pi) = 1.

    Discussed radians being unitless.

    Drew the 5 triangles I want you to memorize, for 0, 30, 45, 60, and 90 degrees (and their Pi-radian versions). With those 5 in Quadrant 1, you can find trig functions of any angle, with terminus in any quadrant, via the REFERENCE ANGLE.

• 2/5 Notes
• 2/5 Video
  • We covered a lot of topics, relatively informally, related to Week 3 Written Assignment.

    Reference Angles, Solving trig equations, drawing the 2 pictures for a trig equation, finding the OTHER solution (The calculator only "sees" one of them.

    I never mess with inverse cosecant, secant, or cotangent. I just always turn statements about them into statements about sine, cosine, or tangent, and I know how to solve sine, cosine, and tangent questions.

    We didn't do a tangent question. Just a sine and a cosine. Ask me about tangent, next time, if you want to see it. Ordinarily, I try to avoid talking all hour. *sigh* Students like when I do, but that's because of years of training for American-style classes, when we're trying to run a "flipped" class.

    It's not that I don't love to talk all hour. It's that everything is pretty much covered in the homework videos, so why bother you twice? Not the best use of your time, to my mind.

    I'd get what I could from the homework videos and (especially the) notes. Use the face time with teacher to ask questions that the on-demand resources didn't quite do well enough for you. That way, ALL your math time is "on-demand" time.

• 2/10 Notes
• 2/10 Video
  • We did some cosine and sine modeling. Discussed the general graphs of sine and cosine.

    We talked a bit about inverses and the IMPORTANCE of drawing pictures for every trig situation.

• 2/12 Notes
• 2/12 Video
  • We covered inverses of sine, cosine, and tangent. We graphed inverse sine and inverse tangent. We graphed cosecant and cotangent, bootstrapping off graphs of sine and tangent, respectively.

    We talked about the restrictions placed on the domains of sine, cosine, and tangent to keep them 1-to-1, and hence make their inverses functions that spit out one number for a given trig ratio. The key to using the calculator is knowing what the picture looks like, so DRAW THE PICTURE!!!

    Domain of the function is range of the inverse
    Range of the function is domain of the inverse.

    We talked about interpreting the mindless output of your calculator when solving a trig equation.

    We talked about standard math shorthand.

• 2/17 Notes
• 2/17 Video
  • Today was optional lecture day, due to Quiz 1, to give people a little breathing room. I prepped a lot of different things, like "How I score assignments" that didn't get covered. Check out page 1 of today's notes for some insight. If you've been turning in your written work, then you already have a good idea what I'm looking for.

    We worked a couple of application problems, today. I drew a lot of triangles, which is what I want you to start doing, now that you're recovering from Section 1.2, which is all unit circle stuff and more complicated than it needs to be.

    I copy-pasted all the questions from Weekly 1 and 2 into today's notes. I used very little of it. But we did get to look at a couple of the more complicated homework questions, one of which I flubbed towards the end, as Alexis pointed out to me after class. I dropped the "35" in the numerator (Lots of "35"s running around in this exercise.). Anyway, I fixed the notes and uploaded them.

    Thanks, Alexis!

    People who missed today's class: Just scroll through all the extra stuff in the middle. The two application questions are near the end.

    Still trying to teach people to use the resources I slaved over, to make their learning more efficient.

    We did the "Where is the fire?" question and the "Build a cosine function" question. I also built an equivalent "Build a sine function" question. You just have manage where to "start" to get the horizontal shift correct. Sine starts at its midline. Cosine starts at its maximum.

• 2/19 Notes
• 2/19 Video
  • I shared a Calculus I student's take on my "5 triangles" concept. He had the lights go on when he saw what I was doing, and tore the dratted 12-point unit circle off of his wall. Unit circle is all well and good, but you want to think in right triangles in different quadrants whenever you're given an angle or a trig ratio.

    Cofunction Identities - a graphical "proof" of one of them.

    Reciprocal Identities, Pythagorean Identities, ...

    Factoring trigonometric polynomials. Gave the class a quick intro to the Sledgehammer method of factoring, using the Quadratic Formula.

    Trigonometric Substitution revisited. It's not hard if you draw triangles. It's very hard if you never draw pictures.

    The most frustrating thing for me when I teach trig is all the students who try to do the work without drawing any pictures. If you don't draw pictures, you don't see it. If you don't see it, you can't really do it for real. You can do some of it, but a lot of it will just fly right over your head.

• 2/24 Notes
• 2/24 Video
  • We discussed Sections 2.2 (Proving Identities) and 2.3 (Solving Trig Equations). We'll do some 2.3 Wednesday and wrap up with Section 2.4 (Sum and Difference Formulas). I don't teach the difference formulas. I teach the sum formulas. You get the difference formulas from the Sum Formulas. Just replace "y" by "-y" in sin(x + y) and you get sin(x + (-y)) = sin(x - y).

    From sin(x + y) = sin(x)cos(y) + sin(y)cos(x), we obtain
    sin(x - y) = sin(x)cos(-y) + sin(-y)cos(x) = sin(x)cos(y) - sin(y)cos(x) = Difference Formula. So just remember the sum formula and also remember that sine is odd and cosine is even.

• 2/26 Notes
• 2/26 Video
  • We discussed Angle Sum formulas from 2.4 and dipped into Double-Angle and Half-Angle formulas from 2.5 (next week).
• 3/3 Notes
• 3/3 Video
  • We discussed 2.5. Emphasis on Double-Angle, Half-Angle, and Power-Reducing Formulas. I gave credit to the sum-to-product and product-to-sum formulas, although I don't emphasize them on tests. You WILL need the product-to-sum formula in Calculus II, to evaluate some otherwise intractable integrals, but I haven't seen a ton of problems that need the sum-to-product formula, even though it is very useful for something like sin(3x) + sin(5x) = 0.
• 3/5 Notes
• 3/5 Video
  • We worked a few different exercises and announced the Written Midterm. We saw two different versions of 1/2-angle for tangent. Both versions are correct, but one of them presents more, uh, presentably.

    I answered (set up) one exercise from Section 3.1 - Law of Sines.

• 3/10 Notes
• 3/10 Video
  • We began Chapter 3, with a brief introduction to Section 3.1 - Law of Sines. I think it got a fair treatment. Then I stated the Law of Cosines from Section 3.2, without doing any examples. Finally, I sprayed a bunch of 3.3 - Vectors in the Plane nonsense at everybody. I hope everyone makes good progress this week and is ready for 3.2 and 3.3 on Wednesday. If you can wrap your head around those 2 sections heading into Spring Break, then you will be in good shape for when we get back.
• 3/12 Notes
• 3/12 Video
  • I did a very quick overview of 3.3 and 3.3 Applications. We covered addition and subtraction of vectors, scalar multiplication, magnitude, and unit vectors. We learned that a vector can be expressed as the scalar product of its magnitude and a unit vector in the same direction. We used this in an application involving tension in two cables suspending a fixed weight (Statics: All forces are perfectly balanced.).

    We saw a visual representation of the resultant obtained from the sum of two vectors: For u + v, you place the butt of v against the tip of u and then form the vector from the origin to the tip of v.

    For the difference of two vectors, u - v is the vector from the tip of v to the tip of u. u - v always points at u.

• 3/24 Notes
• 3/24 Video
  • I need to leave my phone in the other room. Usually there are no interruptions. My big sister retired last week and thought she was missing my class meetings. I thought maybe there was some kind of emergency or something.

    Today was a crash course in dot product of vectors, projection of vectors, and force-times-distance = work (with vectors). This involves the marriage made in Heaven between cosine and dot products.

    We also worked a problem involving a weight suspended from two points, where we found the tension in each cable. This was interrupted by my sister's phone call. For some reason, when she called and I saw it was 9:21, I thought I had gone overtime. It really threw me off. Class meets from 8:15 to 9:30, not 9:20.

    That's more like my calculus class, which DOES meet for an hour and 5 minutes. *sigh*

    Your Week 6's are graded. Your week 7's and Week 8's will be graded by Wednesday.

    We're pretty much done with new material that might be on the Midterm. Next time, we'll review for the Midterm, which is scheduled for Monday or Tuesday of next week.

• 3/26 NotesI left out "Photo ID" for what to bring to Midterm.
• 3/26 Video
  • I was asked about the suspended weight question from 3.3 Applications. I was really snarky towards the WebAssign video and made a boast about "covering everything." I DO have everything covered, but I clearly didn't have all of my videos on these questions posted, so what I said at the beginning of class was inappropriate.

    I feel like the WebAssign videos are for people who already know how to work the problem, because they're doing just enough to convince someone above them, rather than seeming to be talking to someone at their level or below. It may be that I took some of my old videos on this down, because I didn't like them, but then never uploaded the new videos.

    Hopefully, today's video was helpful for the hanging-weight question in 3.3 Apps...

    Let's see. We graphed a cosine function. We didn't build one. You'd have to look above on this page to see where we build cosine functions for things like tides, hours of daylight, and average temperatures. I think you had at least one oscillating spring exercise on the homework. Just don't expect anything too fancy.

    On the half-angle questions, I think people need to draw triangles (2) and select the 1 that fits the description of u. I need to be convinced that you know what you're doing and what quadrant you must be in, and deduce whether it's "+" or "-" on those.

    I solved a trig equation that was more full-bodied than the one on Week 7 Assignment. I took a LONG time to grade the first paper, because I miscopied it on my work, and I knew that the student couldn't be right, but he was right. I sure couldn't find his mistake, but I just had a wicked blind spot on my own mistake. I have all but 2 of the Week 7s done, now, and I will grade those two people's Week 8s first, later tonight or tomorrow morning. I thought I would have 7 & 8 done by now, for everybody, but I didn't have as much time as I planned, due to fighting the Internet, yesterday.

    The 3.4 is due Saturday, 3/29. It's fair to ask a question or three from 3.4. I won't make you find all of the interior angles of a triangle with given vertices, but I will ask you to use vectors to find one of the interior angles of a triangle using vectors and dot products.

    The last stuff I covered in 3.4, about projections and the "Gram-Schmidt" thing, where, given two vectors ubar and vbar, you write ubar as the sum w1bar + w2bar, where w1bar is parallel to vbar and w2bar is orthogonal to vbar, ... Yeah that stuff. Well, I think that stuff is fair game for bonus, for sure, on the Midterm. I intend to allow over-100% grades, so if you're hurting in any other category, performing exceptionally well on the written Midterm can make up for a lot of sins.

• 4/2 Notes
• 4/2 Video
  • We began with a discussion/review of Complex Numbers. I probably wasted too much time on the formal definition of a field and how the complex number field is every bit as much of a field as the real numbers. Add or subtract, you get a complex number. Multiply or divide, you get a complex number. Any complex # that isn't zero has both an additive inverse (The negative of that number) and a multiplicative inverse (The reciprocal of that number). The only way the product of two complex numbers is zero is if one or both of them must be zero (principle of zero products).

    The additive identity for the complex numbers is the real number 0. The multiplicative identity is the real number 1.

    Fact: The complex number field is the smallest field containing the real numbers that is algebraically closed. What that means is that it's the smallest field such that ALL the zeros of ALL polynomials of degree n have exactly n zeros in the complex field. Every nth-degree polynomial has n zeros and splits into linear factors with complex coefficients.

    Translation: The real numbers are not algebraically closed. There are irreducible polynomials that have no real zeros.

    The complex numbers also have some vector-space properties. There's also the analogy of "unit vectors" but we don't really have a dot product.

    I showed, briefly, how multiplication by a complex number of modulus 1 gives a ROTATION about the origin in the complex plane. The amount of rotation is precisely the angle that a line segment from the origin to the number makes if you measure counterclockwise from the positive real (x-) axis.

    Just like (well, almost) the Chapter 3 hanging-weight problems, we can write a complex number in trigonometric form: Vector:

    u = || u || < cos(v), sin(v) >, where v = the direction angle for u.

    Complex number in trigonometric form:

    z = |z|(cos(v) + i sin(v)), where v = the argument of z, Arg(z),

    which is any angle coterminal with its "direction angle," only we don't say "direction angle" when we're talking about complex numbers in trigonometric form.

    Not all of the lecture falls under 4.1, 4.2 or 4.3, or even 4.1 - 4.3. Trigonometric Form of a complex number comes later, as does DeMoire's Theorem, which ties everything together.

    Fact: There are applications to this theory in computer graphics. You want an object to rotate by an angle v? Multiply it by a unit-length complex number whose argument (angle) is v!

• 4/7 Notes
• 4/7 Video
  • I keep "circling back" to what we know about vectors, because so much of it relates to complex numbers. We delved into trigonometric form of a complex number. We went back and forth between rectangular form and trig form.

    We discussed DeMoivre's Theorem, which boils down to

    z1 = r1(cos(u) + i sin(u)) and z2 = r2(cos(v) + i sin(v)) implies the product,

    w = (z1)(z2) = (r1)(r2)(cos(u + v) + i sin(u + v)).

    We multiply moduli and add the arguments!

    This will lead us to nth roots of unity, which is kind of the capstone to the chapter.

    We will discuss the nth roots of a complex number, which is our big application of nth roots of unity.

    I don't know why the completing-the-square part of today's talk switched from x^2 - 2x + 2 to x^2 + 2x + 2. But the work is very similar, whether you're completing the square or using the quadratic formula for both functions.

• 4/9 Notes
• 4/9 Video
• 4/9 Maple Output - Showing how all 5 of the 5th roots we computed for z = 32 (cos(Pi/3) + i sin(Pi/3)) are actually 5th roots of z!
  • We did some housekeeping, today, pointing out how relaxed the home stretch is. Start early and hit it often, say I. The Chapter 6 stuff is pretty challenging, but isn't that bad if you look at it multiple times and build some muscles for it.

    I briefly showed multiplication of (x - (a + bi))(x - (a - bi)) = (x - a - bi)(x - a + bi). Some people have an easier time FOILing the first product. Some have an easier time just distributing each term in the first factor over each term in the second factor. I don't care which you prefer, as long as you're accurate.

• 4/14 Notes
• 4/14 Video
  • Got our Midterms back, today. I ran over some fundamental stuff that will probably be on the final. Anything on the Midterm is fair game for the Final.

    I also worked a square root of a complex # problem.

    As always, I want the class to draw pictures and the class doesn't want to draw pictures. It goes quickly if you can draw the pictures. It's a lot harder to do anything if you're resisting drawing the dad-gum pictures.

• 4/16 Notes
• 4/16 Video
  • Sam and Brody: I apologize for my behavior, today.

    I think you guys GET what a flipped course is and how it works for you. I built the video archive so that you guys would have my answer to virtually every exercise, on demand. Once the archive is built, it's better for you guys if you can access everything you want, when you want it, rather than stopping to listen to me, live.

    I think the lecture should also be on-demand. I just can't seem to get it through my head that people don't have questions, and it's OK if they're working, quietly. Terrible teaching. It comes from a good place, but I'm really struggling to handle people working and me not talking..

    If I'm not getting anything from you during lecture, I will assume all is well, and you will only chat me up if you have a question.

    Just remember that the chat doesn't make any sound. If I don't see your chat question, right away, simply LEAVE and RE-ENTER the ZOOM session. When you leave, it'll ring a bell for me. Then I'll look at the chat and see your question.

    "Who left? Oh. It was Alicia. Oh. Alicia just asked me a question."

    I get why it doesn't ring a bell for chats. If it's an active chat, the bell will be ringing all the time, and make it really hard for the speaker to speak.

• 4/21 Notes
• 4/21 Video
  • Week 13 Written Assignment is posted.

    We reviewed DeMoivre's Theorem in Section 4.5 and took a look at Chapter 6, Section 6.6.

    People are either finished with Chapter 4 or close to finished. This week is pretty much wrapping up Chapter 4 and getting into 6.6 and 6.7.

• 4/23 Notes
• 4/23 Video
• 4/28 Notes
• 4/28 Video
• 5/5 Notes
• 5/5 Video
• 5/7 Notes
• 5/7 Video
• 5/12 Notes
• 5/12 Video

While I don't require your undivided attention during lecture, I do hold everyone accountable for anything that appears in the lecture notes, as I consider the lecure notes a rich resource for Weekly Written Work and especially Written Midterm and Written Final. questions.

I don't think it's too hard to keep yourself apprised of what we're doing in class. Just check in, open the day's notes, and see what was covered. If need be, you can then open up the video for that day, and scroll ahead to the point in the video where I talked about what I was putting on the board.

That said, if you devote yourself assiduously to doing the work, and doing the work in such a way that you're learning the stuff, and getting all your questions answered, you might not even have to check the notes in order to earn a good grade. But there will be Easter eggs sprinkled throughout, so I'd advise at least skimming each day's notes. That may not take you more than 15 or 20 minutes per week, while you spend the rest of your time actually working problems.

The #1 and #2 strongest correlates with a high grade are:

1. The number of exercises you do over the course of the semester; and,
2. The amount of time you spend working exercises and practicing, especially practicing "without a net," in other words, to perform well on tests, you need to practice working exercises without opening your book or referring to your notes.