This week I only got through one unit, Trigonometry With Right Angles. I would say ~80% of it was review, which I found fairly helpful, and the last ~20% of it was new material I haven’t covered up to this point. This week I went away to a cottage and managed to spend about an hour each day working on KA but I likely wasn’t as productive as I would have been if I had stayed home. I’m not too disappointed, however, as I still think it’s likely I’ll get through this course quickly since there’s only one unit remaining that’s not 100% complete. There are three units that are finished that I plan on going through to review , but I’m hoping they won’t take too long.

I heard this past week that the rate of new Covid-19 cases is still increasing by ~1.5% every day which is much better than before but it still seems like we’re a ways away from being on top of it. Considering this, it’s concerning to me that a number of different businesses opened in the city this week. My work is also starting up tomorrow, though the club will only be offering limited services to the members. I’m looking forward to going in and seeing everyone but am uncertain if it’s the right call to open back up. That said, part of me thinks that at some point we all have to just give it a shot in small incremental steps and see what happens. I won’t be doing too much when I’m at work which I assume will give me lots of free time to work on KA, so that’s an extra side benefit!

This week I started to understand the relationship between Sine, Cosine, and Tangent and their related SOH CAH TOA equations much better. I think having learned about the Unit Circle helped me understand these concepts much better. I found it useful practicing questions where I was given a non-90 degree angle of a right triangle and one side and had to solve for the other side using SOH CAH TOA. An example equation of this would be:

• Cos(50 degrees) = Adj/6
  • (6)Cos(50 degrees) = Adj
  • Adj = ~3.857

I also found it useful practicing questions that gave me two sides and asked me to find the value of an angle using the inverse Sine, Cosine, and Tangent functions. For example:

• Tan(Theta) = 35/65
  • (Theta) =Tan^-1(35/65)
  • (Theta) = ~28.3 degrees

 I was reintroduced to the following three terms which I somewhat remembered but has happy to review them and found it very useful. They were:

• Angle of Elevation
  • An angle created by rotating upward from a horizontal line.
• Angle of Depression
  • The opposite – an angle created by rotating downwards from a horizontal line.
• Complimentary Angle
  • Any two angles that add up to 90 degrees. They are referred to as each other’s compliment.

I’m not sure if I had learned this before, but I watched videos and went through exercises that explained that Sin(Theta) = Cos(90 – Theta). This concept makes more sense to me when I remember that it refers to Sin(Theta) being taken from one non-90 degree corner of the right-triangle and Cos(90 – Theta) being taken from the other non-90 degree corner of the right triangle. This makes sense since Sin = Opp/Hyp and Cos = Adj/Hyp and since they’re in opposite corners, Sin’s Opp IS Cos’s Adj.

In my last post I talked about how revisiting certain subjects that I found tricky the first time around often seem easier to understand the second time around. This is exactly what happened this week when I worked on 45-45 triangles and 30-60-90 triangles. These are considered special triangles that have corners with two 45 degree angles and 30, 60, and 90 degrees, respectively.

In a 45-45 triangle, the Opposite side and Adjacent side are the same length, ex. x, and therefore, by using the Pythagorean Theorem, it’s fairly simple to figure out that the Hypotenuse equals x{2}. Figuring out the ratios of the lengths of the sides on a 30-60-90 triangle is more difficult to explain without being able to draw a triangle. The steps to go through to figure out the ratio of the lengths between the three different sides are as follows:

1. Draw a equilateral triangle and state that all sides equal length x. Each angle is therefore 60 degrees.
2. Split the triangle down the middle to create two mirrored, congruent 30-60-90 triangles. By doing so, it’s easy to see that the length of the side between the 60 degree angle and the 90 degree angle equals x/2.
3. Using the Pythagorean Theorem inputting that one side equals x (the Hypotenuse) and one side equals x/2, you can then determine that the remaining side equals {3}x/2.

If the hypotenuse equals x, the values of SOH CAH TOA for an angle taken from a corner equal to 30, 45, or 60 degrees are as follows:

	Cos(Theta)	Sin(Theta)	Tan(Theta)
Theta = 30 degrees	{3}/2	1/2	1/{3} = {3}/3
Theta = 45 degrees	½ = {2}/2	½ = {2}/2	1
Theta = = 60 degrees	½	{3}/2	{3}

The final thing I worked on in this unit, which was something I had not learned up to this point, was the definitions of Cosecant, Secant, and Cotangent. In my mind, I think of these as the reciprocals of Sine, Cosine, and Tangent respectively.  Their equations are as follows:

• Sin(Theta) = Opp/Hyp
  • Cosecant = Csc(Theta) = Hyp/Opp
• Cos(Theta) = Adj/Hyp
  • Secant = Sec(Theta) = Hyp/Adj
• Tan(Theta) = Opp/Adj
  • Cotangent = Cot(Theta) = Adj/Opp

The videos and exercises that I worked through on Cosecant, Secant, and Cotangent were all quite straightforward. They simply asked me to find the value of an unknown side or angle of a right triangle where I was given the other side(s) or angle by using the same methods as I would for Sine, Cosine, and Tangent, but instead by using the Cosecant, Secant, and Cotangent formulas. Right now I don’t really understand why it’s important to know these reciprocal terms/definitions/formulas but I’m happy that I found them simple to understand.

My goal is still to get through this course by the end of June. I think that should be fairly easy since, as I said, I only have one more unit remaining that’s unfinished. Again, as I said earlier, I plan on reviewing the other units (i.e. watching the videos but not working through the exercises or unit tests) but there are only 3 of them and 2 are very short. I feel like I’ve been slacking a bit lately on KA but I’m genuinely excited to get through this course and start the following course Statistics and Probability so I’m hoping I can turn things around and start getting more work done each week. It would be great to review all 3 units this week which I think is possible. We’ll see!