Mission accomplished. Not only did get through the unit Trigonometric Equations and Identities this week, but I also finished the course test scoring 29/30 on my first attempt! If I had taken bit more time to think it through, I probably shouldn’t have missed the one question I got wrong either. There were times in the past two months that I thought I could have potentially gotten through this course quicker, but regardless I’m quite pleased with my progress and pumped to be getting into the next subject, statistics.

(Things are more-or-less the same with Covid so I won’t bother with an update this week.)

All I had left to do to finish the unit Trigonometric Equations and Identities (apart from the unit test) was watch 7 videos in the section Challenging Trigonometry Problems. The word ‘Challenging’ I realized was a pretty big understatement. Each video started with Sal reading a question that involved Trig identities and at first glance, of the 7 videos, I initially understood 0/7 of them. Even as Sal worked through each question, I was only able to fully understand how he got to his answer in 3/7 of them. Being that this was the end of the course, I assumed I had a pretty strong grasp on Trig but, based on these ‘Challenging‘ questions, now I feel like my understanding of trig at this point may really only be scratching the surface.

To be fair to myself, most of the questions in these videos used math that I’m quite certain I’ve never seen before. At one point Sal said he needed to use a bit of calculus to work through the question which I have never be exposed to before this. (I was so lost, I honestly couldn’t tell whether or where he even used it.) Another time Sal said he was using ‘derivatives’ which I also am sure I’ve never used before and was thoroughly confused as to what was happening. Nonetheless, I watched all 7 videos from start to finish and tried my best to keep up with what he was talking about and was able to somewhat understand what was happening.

I had one breakthrough moment this week when I was working through the unit test. It came to me as I was working through questions that asked me to use an Angle Sum Identity. Before that point, I struggled to conceptually understand how to add special triangle side values to each other when the triangles were created using angles -90 > Theta > 90. What I learned this week that helped me to understand how it works is that, wherever an angle is on the unit circle, you draw a line straight from where the terminal line intersects the circumference of the unit circle down/up to the x axis which creates a right triangle. From there you can then label the sides using the appropriate values from 45-45- triangles or 30-60-90 triangles assuming they’re applicable. Take this page out of my notes as an example:

In the above example, I use the Sine Angle Addition Identity to find the Sine of 135 degrees + 300 degrees. As you can see, the 135 degree angle sits in the 2^nd quadrant and can be thought of as a 45-45 triangle flipped over the Y-axis making Cos(135) = -{2}/2 and Sin(135) = {2}/2. The 300 degree angle goes all the way around into the 4^th quadrant and can be thought of as a 30-60-90 triangle flipped over the X-axis making Cos(300) = ½ and Sin(300) = -{3}/2.

Considering how difficult I found the questions in the ‘challenge’ videos to be, I was pretty shocked at how easy the course test was. The test didn’t have any questions that were even remotely as difficult as the challenge questions were. Most of the questions on the test went back to sinusoidal graphs/equations, had me solving a Sine or Cosine equation (i.e. figuring out the degree/radian measures that aligned with the equation), or adding/subtracting angles using the Sine/Cosine/Tangent addition/subtraction identities. Like usual I was nervous as I hit enter for each of my answers throughout the test but, unlike during past course tests, this time I felt more confident in my understanding of the material which was super satisfying.

As I mentioned earlier, I could/should have been able to answer the one question I got wrong correctly. It was a word question that asked me to come up with a sinusoidal equation to represent the annual temperature fluctuations of a country which I can’t remember specifically. The end of the question stated something along the lines of, “write the equation to represent the temperature fluctuations over multiple years” which essentially asked me to set the period of the equation to equal one year. I have been given many questions like this before and every single one in the past asked me to make the period equal 365 days which is exactly what I did which is why I got the question wrong. Here’s a picture of of my notes that shows how I answered it and why I got it wrong:

I was bummed that I got it wrong but was happy knowing that I completely understood why I got it wrong and that I probably should have gotten it right if I would have thought through it a bit more. I was also pumped that I got the rest of the 29 questions correct. 🙂

I’ve said this about 100 times throughout the last few posts, but I’m fired up to finally be getting into statistics. I think I’m more excited than usual because it’s the last subject before calculus which is the subject I’ve been wanting to study since the very beginning of this endeavour. There are three statics courses for me to get through, however, before I begin calculus, two of which are quite large and the other also being a decent size. In chronological order, the courses are Statistics and Probability, High School Statistics, and AP® Statistics. I’m hopeful and think it’s likely that the initial few videos and exercises in the first course Statistics and Probability won’t be too difficult. I have a feeling I’ll be going through different graphs and charts which I would have learned about in high school. If that’s the case, will hopefully I’ll be able to recall a lot of the information and will initially find it all straightforward and simple. In any case, time to finally start stats!