So I didn’t finish the unit Trigonometric Equations and Identities this week. My worry that the unit would end up being substantially more difficult than the previous ones in Trigonometry ended up coming true. I also came to a section that I’m fairly certain KA hadn’t properly prepared me for. That being said, I did get make some solid progress this week which I’m happy about and even started to figure out the section that I was under prepared for. Looking ahead, with only two weeks left in the month, I realized there’s a chance I may not get through the course by the end of June which would be disappointing not to hit my goal.

I ended up taking a Covid test this week and the results came back negative. I got the test because for the past 9-10 days I’ve had an irritated throatand thought it would be a good idea to check. In a weird way, there’s a small part of me that wishes I had it so that I would (presumably) get over it, develop antibodies and not be so stressed out about the whole thing. The more rational side of me is happy with the negative result though. As well, positive test rates have been dropping across Ontario which is a hopeful sign!

First off, I made a mistake last week on the restraints used on Arcsine. I said that the restraints went up to Pi/2 and Pi/2 (a.k.a. 90 degrees and negative 90 degrees) but didn’t include them but it turns out they are included.

WRONG	RIGHT
Arcsine	Arcsine
Pi/2 < Theta < -Pi/2	Pi/2 ≤ Theta ≤ -Pi/2

After I realized my mistake from last week, the week began by working through questions where I was asked to ‘solve’ sinusoidal equations. ‘Solving’ a sinusoidal equation basically means to figure out a given angle on a unit circle and state that (the angle) + n*2Pi (where n = any integer, positive or negative) are all possible solutions to ‘solve’ the equation. The previous equation means that for any given angle, adding 360 degrees to it in either the positive or negative direction will result in the same angle, a.k.a. the same solution. An example of one of these types of questions I was asked to solve would be:

• (8)cos(12x) + 4 = -4
  • (8)cos(12x) = -8
  • Cos(12x) = (-8)/8
  • Cos(12x) = -1
  • 12x = Cos^-1(-1)
  • 12x = 180 degrees + (n*360 degrees)
    • On the step directly above, I learned to insert (n*360 degrees) which basically states that, “the solution is 180 degrees plus any positive or negative multiple of 360 degrees.
  • 12x/12 = 180/12 degrees + (n*360)/12
  • X = 15 degrees + n*30 degrees

To be honest, I don’t exactly understand what the final statement X = 15 degrees + n*30 degrees means. My assumption is that it means for the initial equation, the primary solution is x = 15 degrees but adding any positive or negative multiple of 30degrees to 15 degrees is also a solution.

One thing I realized this week is that Sal often switches from radians to degrees from video to video and doesn’t always outright say which unit he’s working with. I realized this must have been very confusing to me when I was first introduced to radians. I now have a solid, intuitive understanding of when he’s using which unit based on the numbers he associates with the angle which has made all of these concepts much easier to understand and follow along with.

The most difficult part of the week (which as I said, I felt thoroughly under prepared for) came when I got to a section called Introduction to the Trigonometric Angle Addition Identities. There are 5 videos in this section, all of which I’ve watched twice through at this point. After initially feeling completely lost, I realized I need to go back a few units and re-watch Sine, Cosine, and Tangent Symmetry Identity videos to better understand what was happening. The Symmetry Identities are another thing I find difficult to explain in words but I’ve inserted a photo below of two pages of notes that sum up how the identities work. It may be difficult to understand just by looking at the photo, but these notes explain how the symmetry between the Sine and Cosine of an angle Theta in angle “A” relate to angles “B”, “C”, and “D” by flipping that theta over the X- and Y-axis’s.

After going back and re-watching the basic Sine, Cosine and Tangent Symmetry Identity videos and getting a better understanding of how those identities work, I was able to somewhat figure out two more identities for Sine and Cosine that deal with adding two angles. Those identities are:

• The Sine Angle Addition Identity
  • Sin(x + y) = (Sin(x))(cos(y)) + (cos(x))(sin(y))
• The Cosine Angle Addition Identity
  • Cos(x + y) = (cos(x))(cos(y)) – (sin(x))(sin(y))

I really don’t have a strong understanding of these equations but they both essentially state, if you stack two right triangles on top of each other you create a third triangle (which you can think of as as being layered on top of the first two) and, by using the above identities, you can solve for the third triangles Sine and Cosine values. (Confused?) I’ve barely wrapped my head around how these identities work and am sure I’m going to need to re-watch these videos a few more times to get a better understanding of why they work. It’s all slowly starting to make sense though which is motivating.

There are 15 days left in June. I have 11 more videos to watch and 2 exercises left in this unit plus the unit test. I’ll also need to do the course test. I’m reminding my self that my main goal is to understand the material so if I don’t end up finishing the course by the end of the month because I’m taking my time to absorb and understand the concepts, it won’t be the end of the world. Nonetheless, I definitely want to make a big effort to get it done by the end of the month which I still think is doable.

Time to get back to work!