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.
We had 22 minutes remaining, so I dove into Chapter 4: Complex Numbers. We did some basic calculations and I tried to give you a sense of how multiplication of complex numbers amounts to multiplying their magnitudes together and adding their angles.
Referring back to "vector = magnitude of vector times unit vector in the right direction," I gave you a glimpse of "Trigonometric Form of Complex Numbers," which looks very similar. If magnitude of z1 is | z1 | and its "direction angle" is u, and magnitude of z2 is | z2 | and its "direction angle" is v, then
z1 = | z1 | (cos(u) + i sin(u)) and z2 = | z2 | (cos(v) + i sin(v)) and the product (z1)(z2) is given by
(z1)(z2) = | z1 | | z2 | (cos(u + v) + i sin(u + v).
Then I worked Section 3.3 Applications hanging-weight, where we had lengths but no angles. I showed how my OCD simplifications can be a waste of time, if done prematurely. Sometimes a timely simplifaction can reduce the work involved. Sometimes it just sucks up time, to no purpose.
Speaking of wasting time, if we can con Judah into sharing his test with us, you can see how he left some of the more tedious simplifications unfinished. He still got 4.5/5 for most of those, because he did all the steps, except that last (OCD) simplification.
I wanted to point out that you never want to spend more than 5 minutes per problem on your first pass through the test. If you're not totally finished, start the next problem on a new page. Frequently, everything you do in the first 5 minutes is going to be most of the points you earn for the problem, anyway! And worrying the problem to death trying to make the answer perfect will mean you didn't have time to do all the other problems. If you get to the end, with time left over, you can always go back to previous problems.
Another good thing about this strategy is that your subconscious works on all of the problems "behind the scenes" and you get ideas on how to work problems you passed on the first time through.
At the test, they tell you to use both sides of the blank paper provided. I recommend AGAINST that. It's way easier to keep the stack straight and in proper order of problems if you only write on one side.
Good work on the Midterm, btw. Thanks, Yasmin, for your generosity in sharing some of your stuff.
I tried to derive the dot-product formulation for the cosine of the angle between two vectors. Everything went well, until I confused ||u|| = sqrt(u*u). I left off the "sqrt" part in one piece of the derivation. I patched it up, but I'm not very happy about it.
That pretty much wipes out all the highlights of Chapter 3. It remains to nail down all the details and all the assignments.
Today, we (haltingly) discussed Section 3.2 and I worked #12. It looks like there's no #12 in the videos and I'm not sure, but I think that the notes are incorrect for #12!
I provide a partial proof and work some examples.
There are 3 possibilities for one of these problems:
We talked about the Midterm: Monday, October 13th OR Tuesday, October 14th, Horizon Hall (HOR) 107. Bring ID, scientific calculator and pen or pencil.
Show up any time any of those days, between 8 am and 6 pm to take your midterm.
You are permitted a one-page (both sides) cheat sheet with whatever formulas you want on it. I recommend AGAINST putting examples of particular exercises on there. Students who do that rarely seem to succeed on tests. You want to know how to work all the type problems, and your cheat sheet should just be the identities and formulas.
I shared a link to a cheat sheet.
We talked about my 5 triangles vs the 12-point unit circle.
I had a question from Dany that I didn't catch. Sorry I lost you, today, Dany! I was just too keen on covering the graphing. If you already did your 1.5, then you probably already saw everything I said, today, in the homework notes and videos, which give pretty comprehensive coverage of all the skills.
In direct response to your comment, yes. I encourage you to use graphing utilities to help you visualize these functions, especially the cosecant, secant and cotangent. At the same time, I want you to be able to do as much as you can with just your brain, pencil, and paper. Next Monday, I will try to show how to get graphs of cosecant, secant, and cotangent from the graphs of sine, cosine, and tangent, respectively.
I think you were 1 lecture ahead, and I didn't honor it very well. I could easily have demonstrated using DESMOS and/or WOLFRAMALPHA, for quick "intuition-building" visuals. These tech tools can also be a quick check on what you are attempting to master with paper-and-pencil. But eventually, we want you to do as much as possible by hand, with the assistance of a scientific (not graphing) calculator.
To know the limits of what might be asked, please refer to Old Midterms and Finals .
So I want you to thoroughly understand the graphs of sine, cosine, and tangent. The rest can be reasoned out, with the more classic, paper-and-pencil techniques I will stress. It's paper-and-pencil that will be tested on the Midterm and Final, and the paper-and-pencil skills are what you need down the road, if you want to know how to tell a computer to do the work. I caution against being too technology-dependent.
When you're too tech-dependent, you may make incorrect reports because you hit one wrong key, but didn't have the commen sense to realize that your results were ridiculous. "Really? 10 tons of flour to double my cake recipe that calls for 4 cups?"
There's a wealth of material from College Algebra on graphing basic functions and transforming them into more complicated functions' graphs. Check out Section 1.5 Homework Notes and Videos.
Instead, what I do is spread that 15% proportionally across all categories and average over 85 instead of 100, with the remaining categories remaining the same, relative to one another, just without the Quiz category being included.
Today, we talked about some 1.4 questions from WebAssign. 1.4 is when I get out of jail and can show you the main skills, without holding anything back. In the past, I've skipped 1.2, to keep you from trying to memorize the 12-point unit circle, when you only need 3 triangles, 2 degenerate triangles, and logic, to find any of the 12 points of interest around the unit circle. Less memory. More ideas.
We discussed trig functions of any angle, which is Section 1.4. The same ideas that we learned for Quadrant I apply to the other 3 quadrants, except you have to keep straight which (if any) of the x or y are negative.
About all I really use the Unit Circle for is to prove the Pythagorean Identities that are used on the later exercises for 1.3 and the Week 2 Written Assignment.
Remember, you're free to call me, any time, at 970-290-0550.
This isn't for everyone, necessarily, but it will really speed things up and help your visual understanding, compared to memorizing sine and cosine for 12 angles, separately! (36 items versus (really) 3 triangles and your reasoning skills.)
I did NOT show how you'd solve sin(t) = 1/2 in detail. I will next time, so you see that basic trig-equation-solving technique as early as possible, whether it sinks in, or not. If it does sink in, this class will be a snap for you! If it doesn't sink in, right away, you'll just pick it up the next time or the time after. When I learned this stuff, I usually failed at each new thing 2 or 3 or 10 times before it clicked.
I was never all that smart. I just persevered until I got it, even if it took me all night. "You figured that out, fast!"
"Nah. I just stayed up all night. I've spent 12 hours on this since last we spoke."
I showed something of how the website is structured. Bare bones. Basic. Best practices for the 1980s, built to minimize "mean time before finding" (MTBF).
Jordan asked me about 1.1 #24 part (b), and I did a poor job of it, but we arrived at the correct answer by dividing our answer for Part (a) by 3600 (seconds per hour) and multiplying by n. The second part of Part (b), we just replaced 'n' by 't,' as the rate was 1 revolution per second on the front sprocket.
I'm still not happy with my explanation, but on the bright side, I'll only ask you questions on Part (a). It finally clicked in my brain after Jordan had the answer (and quite enough of me!). I'll take a stab at it on Monday, if I remember and/or anyone asks.