This week I achieved my goal of getting through two units, The Unit Circle Definition of Sine, Cosine and Tangent and Graphs of Trigonometric Functions. I finished about half of the former unit in Week 39 so wrapping that up wasn’t too difficult. Getting through the latter unit took me about 5 days but proved to be very useful review and which helped me finally connect the dots on a couple concepts I’ve been struggling with. I also managed get through 5 videos of the final unit of the course, Trigonometric Equations and Identities. I’m happy with the progress I made this week and have been thinking a lot about the fact that I’ll soon be starting the first statistics course which I’m (as a young Will Malmo would say) stoked about.

(There’s no new noteworthy news to report about Covid-19 so I’m not going to do an update paragraph this week.)

I began the week by learning about the Pythagorean Identity which takes the Pythagorean Theorem and applies it to the Unit Circle.

• Pythagorean Theorem:
  • a^2 + b^2 = c^2
• Pythagorean Identity:
  • Opp^2 + Adj^2 = Hyp^2
  • Sin^2(Theta) + Cos^2(Theta) = 1^2
  • y^2 + x^2 = m^2

One thing I learned was how to properly write out and pronounce Sin^2(Theta) and Cos^2(Theta) (before I would have assumed they were written and pronounced Sin(Theta)^2 and Cos(theta)^2). Another thing I learned in this section was that Sal Khan thinks of the angle inside the unit circle as being measured in radians whereas he pictures the circumference of the circle being measured in radii. In the past few posts I’ve mentioned that I figured out that radians measure both the central angle in a circle as well as it’s circumference so, considering how Sal views it, it seems like I may have to slightly adjust the way I think about it.

Next I learned about a new number/value/sign known as Tau which equals 2(Pi). From what I’ve come to understand, Tau was created to be in place of 2(Pi) as it makes a bunch of formulas (especially ones using the Unit Circle) easier to intuitively understand. This is because 360 degrees = 2(Pi) which makes an angle such as 270 degrees = 3(Pi)/2 where as if you used Tau it would simply be 3(Tau)/4. Since 270 degrees is ¾’s of the way around the circle it intuitively makes more sense for most people to think of 3(Tau)/4 as opposed to 3(Pi)/2.

I then got into the unit Graphs of Trigonometric Functions which was a review on sinusoidal functions, recognizing their components, their equations, how to manipulate them through their equations, etc. I had a major breakthrough going through this unit when I better figured out how to extract the period from a sinusoidal equation.

• When looking at a sinusoidal equation, it’s helpful to imagine that everything inside the brackets of the part of the equation that deals with the period has to add up to ~6.28 (i.e. 2(Pi)) and then solve for x. Two examples of how this works would be:
  • y = (5)Sin(2x) + 10
    • (2x) = ~(6.28) = 2(Pi)
    • 2x/2 = 2pi/2
    • x = Pi
    • Therefore, the period of the function y = (5)Sin(2x) + 10 is x = Pi.
  • y = Sin(Pi*x/5)
    • (Pi)x/5 = ~6.28 = 2(Pi)
    • (Pi)(5)x/5(Pi) = (5)2(Pi)/(Pi)
    • x = 10
    • Therefore, the period of the function y = Sin(Pi*x/5) is x = 10

I had another breakthrough getting a better understanding of how to shift a sinusoidal function along the X-axis by either adding or subtracting a number from x in its equation. I realized it’s helpful to draw out a Unit Circle and label the maximum value (i.e. top of the circle), minimum value (bottom of circle), and midpoints (sides) with their corresponding period values from the equation. From there, based on the values of the period I’ve written down on the Unit Circle, it makes it easier for me to understand that if I need to shift the function to the right I subtract that value from x and if I need to shift the function to the left I add that value to x. To be honest however, even though this does help, I still don’t have an intuitive understanding of why this works but I do feel like I’m getting closer to fully understanding it.

As I mentioned, I got through a few videos in the last unit of the course, Trigonometric Equations and Identities, at the end of the week. The 5 videos that I watched introduced me to Arcsine, Arccosine, and Arctangent which, as far as I understand, are the exact same thing as Sin^-1, Cos^-1, and Tan^-1. This is confusing to me since, at this point, I don’t see the reason of having two names for the exact same thing. This makes me think there must be some difference between the two that I just haven’t learned. Regardless, an analogy that helps me understand how Arcsine/Sin^-1 and the lot work is by comparing them to exponents and logarithms. For example:

• 2^3 = 8
  • Log_(2)8 = 3
• Sin(60 degrees) = Opp/Adj = ({3}/2)/1
  • Arcsine(Opp/Adj) = Arcsine({3}/2)/1 = 60 degrees
  • Sin^-1(Opp/Adj) = Sin^-1({3}/2)/1 = 60 degrees

In the same way the logarithms switch the order of the equation, arcsine/Sin^-1 work in a similar way. Arcsine/-cosine/-tangent allows you to find the ratio between any of the three sides by knowing one of the angles.

Finally, the last thing I learned about Arcsine/-cosine/-tangent is that they are restricted to certain values on the Unit Circle. To be honest, I’m not entirely sure how/why this works, but it has to do with their domains and ranges and the fact that when you take the inverse of a function, i.e. mapping from a value on the range (Y-value) to a value on the domain (X-value), there can only be one answer which is why you must restrict (constrain?) the range. Though I don’t completely understand it, the chart I wrote down in my notes to help remember the restrictions is:

Arcsine	Arccosine	Arctangent
Pi/2 < Theta < -Pi/2	Pi ≤ Theta ≤ 0	Pi/2 < Theta < -Pi/2
1st and 4th quadrant	1st and 2nd quadrant	1st and 4th quadrant

It’s worth noting that Arcsine and Arctangent do not include the maximum (Pi/2) and minimum (3(Pi)/2) points of the unit circle but Arccosine does include the midline points (0 and Pi). As I said, however, I still don’t fully understand this concept but I’m hoping that as I continue to go through this section I’ll not only understand this particular concept better but also get a better understanding of how the domain and range of all other functions work, as well.

The final unit, Trigonometric Equations and Identities (0/60 M.P.), only has 6 exercises which compared to other units isn’t too bad, but there are close to 20 videos I’ll need to watch to get through it all which will certainly take a bit of time. My goal this week is to simply finish the unit which, assuming the remaining sections aren’t majorly difficult, shouldn’t be an issue. Assuming I do it, I should then be able to finish the course test by early next week which means I’ll finally be starting statistics next week, too. Looking back at all the work I’ve done to finally be starting stats, I’m pretty proud of how far I’ve come.