This week I only got through a unit and a half on KA. I finished the unit Trigonometry with General Triangles and got halfway through the following unit, The Unit Circle Definition of Sine, Cosine, and Tangent (typing that out, I just realized how terrible or a name that is). I probably should have got more done this week than that, though I’m sure I did more than 5 hours of work in total which from the start has been my minimum goal per week.

In keeping with the last few weeks, Covid-19 restrictions continued to ease over the past seven days. The squash club I work at re-opened and ended up going into work Monday, Tuesday, Thursday, and Friday. The club is only offering a handful of restricted services, however. In some ways it was nice going back into work and seeing some familiar faces but in other ways it was a bit of a bummer, especially having to leave my dog at home! When I think about it, I always knew that at some point things would eventually have to go back to normal and so it’s not the end of the world. It’s also a sign that things with the virus are (hopefully) getting better so, overall, I’m happy that my work has re-opened.

The first unit I worked through, Trigonometry with General Triangles, had 3 parts to it; sections on the Law of Sines, Law of Cosines, and solving general triangles. I had learned about both the Law of Sines and the Law of Cosines before this unit but I couldn’t really remember either of them so I found these sections quite helpful. The Law of Sines essentially states that in a general triangle (i.e. not an equilateral, isosceles, or right triangle), the ratio/fraction of the Sine of an angle and it’s opposite side is constant (a.k.a. the same) between the other 2 angles and their opposite sides. An equation of this concept would be

• Sin(a)/A = Sin(b)/B = Sin(c)/C
  • a, b, and c are the 3 angles inside the triangle and A, B, and C are their respective corresponding opposite sides.
  • To reiterate, the equation states that the Sine of any angle and it’s opposite side is the same ratio/fraction as the Sine of the other two angles and their opposite sides.

I was shown why the Law of Sines works, however it’s difficult to explain here without being able to draw a triangle. It has to do with drawing a perpendicular line (a.k.a. an amplitude) down from one angle to its opposite side to create two inner right-angled triangles, and then by using the SOH and a bit of algebra on the two inner triangles you can prove that Sin(a)/A = Sin(b)/B.

The Law of Cosines is even harder for me to explain without being able to draw a triangle. The Law of Cosines states that the value of any given side when squared is equal to the value of the other two sides squared and added to each other, minus two times the value of each side and times the Cosine of the angle between those two sides (simple, right?). The formula for this is a^2 = b^2 + c^2 – 2bcCos(A) and the side a must be opposite to the angle A. The way to derive this formula is by dividing a given general triangle into two right triangles and using the Pythagorean Theorem and algebra to eventually come to the formula. I really have no idea how to describe it well here without being able to draw a triangle so I’m not going to try. It does bother me, however, that I can’t explain it in words and makes me think that I probably don’t understand it well enough if I can’t do that. (To be fair, however, even the KA review section where they wrote down things to remember about Law of Sines and Law of Cosines, they don’t actually explain through writing how to derive each formula but instead direct you to a video that explains them.)

The unit ended with me working through a section called Solving General Triangles. In the videos I watched, I was shown a triangle with a missing side or angle and had to determine whether to use the Law of Sines or Cosines to figure out the missing value. Overall, after just reviewing both laws, I found this part very simple.

The next unit, The Unit Circle Definition of Sine, Cosine, and Tanget, was for the most part (as the name would suggest) a review on the Unit Circle. The unit began, however, going over radians and how to convert degrees to radians. I found this useful to really strengthen my understanding of how radians work. When I was first introduced to radians I struggled understanding them so it’s immensely satisfying now having confidence in my understanding of how they work. What helps me the most when it comes to radians is remembering that:

1. Radians measure the circumference of a circle AND the angle from the center of the circle.
2. 180 degrees = Pi(radians) which basically states that the circumference of a semi-circle equals ~3.14 times the radius of the circle, AND that 180 degrees is Pi * radians.

The second last thing I got through this week was taking an angle on a unit circle (in the video that explained this, the angle they used was~30 degrees) and flipping it across the X- and Y-axis’s and then stating the resulting angles Sine and Cosine values. This gave me an appreciation and better understanding of the tangent of each of the other three angles created (a.k.a. their Hypotenuses, a.k.a. their slopes) and helped me understand why each respective tangent/slope was either positive or negative.

The final thing I worked on was taking an angle on a unit circle (the example angle was again ~30 degrees), adding 90 degrees to it (a.k.a. Pi(rad)/2) and trying to understand how the Sine and Cosine values were altered by doing so. This was new to me and I still don’t have a great understanding of how/why the values get changed the way they do but what I wrote down in my notebook was:

• Sin(Theta + Pi(rad)/2) = Cos(Theta)
  • This equation states that the value of Sin(Theta + 90 degrees) (i.e. the Y-axis value of the resulting, altered angle) is the same Cos(Theta) value of the original angle (i.e. the X-axis value of the first angle). Therefore, adding Pi(rad)/2 to Theta makes the X-axis value in the first angle become the Y-axis value in the second angle.

As a final note, I realized near the end of the week that this unit helped me to realize that 1) Sine, Cosine, and Tangent, in terms of SOH CAH TOA, only deal with right angles, and that 2) Sin(Theta) and Cos(Theta) will always equal a fraction/decimal less-than 1 since the Opposite and Adjacent sides of any right-angles triangle will always be less-than the hypotenuse. Part of me feels like those are two fairly obvious and simple concepts to understand which I should have realized long before this but, regardless, I’m happy that I understand them now.

I’m starting Week 40 this week which seems like a bit of a milestone and which also happens to be starting on June 1^st. I’m hoping I can get through the remainder of this unit, The Unit Circle Definition of Sine, Cosine, and Tanget, and also finish going through the following unit, Graphs of Trigonometric Functions, by the end of the week. The latter unit is 1100 M.P. so there could be a number of videos I’ll need to review but, if I’m able to get through it this week, that will give me 3 weeks to finish the final unit of the course, Trigonometric Equations and Identities (0/600 M.P.). Assuming I can also get through the course test in those 3 weeks, I will achieve my goal of finishing this course by the end of the month. Woo!!