I made a lot of progress this week and was able to finish off Trigonometry and Complex Numbers from Precalculus. There were only ~12 – 14 videos and a handful of exercises between the two units, but I also had to redo each unit test so it added up to a solid amount of work. I also got started on the unit Matrices making it through 7 videos and 2 exercises. I thought it was going to take me until September to get through the 4 unfinished units in Precalc but at this pace I think I could potentially get through them within a few weeks!
(Hopefully I didn’t just jinx myself.)
I forgot to note how many videos/exercises I needed to get through in Trigonometry but I think it was something like 5-6 vids and 3-4 exercises. For the most part, none of the vids/exercises I went through were too difficult. One video was for the proof of tan(a + b) = (tan(a) + tan(b))/(1 – tan(a)tan(b)) and tan(a – b) = (tan(a) – tan(b))/(1 + tan(a)tan(b)). Here are the notes I took from that vid:
I started the trig unit test late on Tuesday which took me into Wednesday to complete. I can’t remember but I think I passed it on the first try and found everything pretty straightforward except for the following question:
As you can see from my notes compared to the answer given on KA, I used the wrong approach to solve this question. When I got to my solution, I assumed I had it wrong and didn’t actually calculate what the value of my final answer, (–√2 – √18)/8, was equal to. Of the four given answers, my answer looked like it was closest to –√2/2 so I chose that answer and it turned out to be correct. It wasn’t until later in the day that I wondered if (–√2 – √18)/8 was equal to –√2/2 which, as it turns out, it is as they both equal ~(–7.07). Here’s the way I should have solved it:
There was one other question on the unit test that I struggled to solve although I did end up getting it correct in the end. As you can see below, the question was, “What are all the solutions for 7cos(9x) – 1 = 1. After answering the question, I spent some time trying to think though it and had a minor epiphany by working through the question backwards. Here’s the question and my notes as to how/why the solution works the way it does:
I began the unit Complex Numbers on Thursday. There were 6 videos and 2 exercises to get through which took me until Friday to finish. Complex numbers seemed less intimidating to me this time around, although I had a hard time remembering how complex numbers were calculated in what’s known as ‘polar form’. Here’s a (messy) page from my notes that goes through the important definitions of CN’s in polar form:
I was able to remember how to add, subtract, and multiply complex numbers but couldn’t remember how to divide one by the other. Luckily, one of the videos I watched showed me how to do it:
I started and finished the unit test on Friday which was only 10 questions long, but it took me FOUR attempts to get through it… On each attempt I’d make a simple, careless mistake, get a question wrong, and then need to start over again. The test was easy for the most part but it took me a few attempts to remember/wrap my head around the following type of question on polar multiplication:
(⬆️ This might be the worst note I’ve ever added to a post lol.)
The gist of how to solve these types of questions is that you determine what the angle of the multiplier is and then add that angle to the multiplicand, and expand or contract the multiplicand by multiplying it by the value of the modulus of the multiplier. (That was the most math-y sentence I’ve ever written. 🤓) In the question above, the multiplier is 3i which has angle of 90° on the complex plane so z has to be rotated 90°. The modulus of 3i is 3 so z also needed to be 3x the distance it was originally from the origin.
I was happy to get started on the unit Matrices on Saturday. There were a total of 19 videos and 4 exercises I needed to get through in this unit and I’m happy to have gotten through 7 vids and 2 exercises on Saturday alone! The first few videos were very simple and just had me looking at tables of random info and more-or-less add the rows and columns together. After that, I was reintroduced to what’s called a ‘matrix determinant’ which I couldn’t remember much about. I finished Saturday watching a video about how matrix determinants help to find the area of a vector transformation. At this point, however, I don’t have much of a grasp on why it works. In any case, here are two pages of my notes that talk a bit about both concepts:
As I mentioned at the start, I’m now hoping I can get through both Matrices (960/1,200 M.P.) and Probability and Combinatorics (600/1,400 M.P.) in the next few weeks. I have 7 vids and 2 exercises in the former unit and 13 vids and 8 exercises in the latter, plus both unit tests. I’m not sure if I mentioned this in a previous post but I’ve also decided to 1) spend a minimum of 1.5 hours each day on KA and 2) write the entire post on Sunday so that I can spend Monday working on KA instead of just writing the blog post. With the extra half hour per day and spending 1 more day per week working on KA, that puts my minimum amount of time up to 9 hours per week instead of 5 hours. If I can maintain that pace, that’s an extra 80% more time spent working on KA! I’ve been able to do it for the last 2 or 3 weeks so, if I can keep it going, I definitely think I can get through Precalc in a relatively short amount of time. 💪🏼