Once again I did not have the successful week I was hoping for this week and just barely managed to get through one unit, Trigonometry. I would say, however, that I worked harder this week than any other week up to this point and was able wrap my head around certain concepts that before this I really struggled with. It was one of the most frustrating yet most rewarding weeks I’ve had. At this point, after studying trigonometric functions for close to 20 hours over the last 7 days, I’m literally picturing the highs and lows of the week in graphical form… And I’ve never felt like such a nerd.

Corona Update – Canada is still getting ~1700 new cases of the Coronavirus each day but it looks like the curve may be starting to flatten. Total cases around the world are just below 3 million with the U.S. just below 1 million. I feel more calm in general about the entire situation than I did a week ago, but it still feels we’re likely nowhere near the end of this. Bill Gates said that until we get a vaccine (i.e. 11-17 months from now) at best going back to “normal” will still mean wearing face-masks, not shaking hands, no large public gatherings, etc. It’s hard to picture things staying the way things have been for the past month for an entire year but it looks like that may be indeed how this whole thing plays out.

This week’s unit, Trigonometry, was based around what is known as the Unit Circle which can be thought of as a circle on a graph centered around the point (0, 0) with a radius of 1. The Unit Circle is what’s used in trigonometric functions and sinusoidal/periodic graphs. Key things to know about the Unit Circle are:

• The angles measured inside a Unit Circle always start from the line along the X-axis going from (0, 0) to (1, 0). Somewhat related to this, the X-axis is used in trig functions as the adjacent line (more on this below).
• The Terminal Line of an angle drawn on a Unit Circle is the line that goes from the center point (0, 0) through any other point on the circle which creates an angle from the line (0, 0) to (1, 0). The Terminal Line is used as the hypotenuse in trig functions.
• In any right triangle drawn on a Unit Circle, the hypotenuse is the radius of the circle and always equal to 1 unit in length. This makes the adjacent line of the right triangle equal to the x-value and the opposite line created equal to the y-value.
• Since Cos(theta) = Adjacent/Hypotenuse (C.A.H.) and the hypotenuse equal 1, then Cos(theta) = Adjacent = x. Similarly, Sin(theta) = Opposite = y.

After being introduced to the Unit Circle, I then worked through a section on radians. In Algebra 1 I struggled to understand radians but working through them here it all started to click and seemed fairly simple. I realized that a radian is a unit of measurement for two things simultaneously; 1) the length/section of the circumference that equals the length of the radius, and 2) the angle that is created from the center of a circle when measuring the length of 1 radian on the circumference. The idea that radians measure two things at the same time, the angle and the circumference, is what I had a hard time understanding initially. Lastly, a key formula to remember when switching between degrees and radians is 180 degrees = Pi Radians. This is derived from the formula for the circumference which is Circumference = 2(Pi)(r).

Next I learned about the Pythagorean Identity which is essentially the Pythagorean Theorem but used in the context of the Unit Circle. It is used to find the values of an unknown side (i.e. the unknown x- or y-coordinate).

• Pythagorean Theorem
  • a^2 + b^2 = c^2
• Pythagorean Identity
  • adj^2 + opp^2 = hyp^2
  • x^2 + y^2 = 1^2 = 1

It’s worth noting here that when measuring an angle by going counterclockwise around the Unit Circle, the measurement has a positive value and when going clockwise around the Unit Circle, the angle has a negative value. This is similar to the idea that going to the right or upwards on the X- and Y-axis gives you a positive value and going to the left or downwards gives you a negative value.

The following is a chart that indicates the x- and y-values (i.e. the Cos(theta) and Sin(theta) values) of a point on a Unit Circle that are ¼, ½, and ¾’s and fully around the circle:

Distance Around Circle	Radian Value	X-, Y-Coordinate	Sin(theta) Value	Cos(theta) Value
0	0	(1, 0)	0	1
¼	Pi/2	(0, 1)	1	0
½	Pi	(-1, 0)	0	-1
¾	3/2 Pi	(0, -1)	-1	0
1	2 Pi	(1, 0)	0	1

Having learned about the Pythagorean Identity, I was then taught about ‘special angles’ which I found out are worth memorizing to know the corresponding values for Sin(theta) and Cos(theta) (a.k.a. the x- and y-values). The angles are measured off of the X-axis in either the (+) or (-) direction and can go into either the (+) or (-) Y-direction. They are the following:

0 degrees	30 degrees	45 degrees	60 degrees
Sin(0) = 0	Sin(30) = ½	Sin(45) = {2}/2	Sin(60) = {3}/2
Cos(0) = 1	Cos(30) = {3}/2	Cos(45) = {2}/2	Cos(60) = 1/2

When using finding the values for these ‘special angles’ it’s important to pay attention to which quadrant the angle is in in order to give the value the appropriate (+) or (-) sign.

The second half of the unit taught me about sinusoidal/periodic functions. These are functions that look like a uniform wave when graphed that go across the graph from left to right. Sinusoidal functions utilize trigonometry and the Unit Circle to describe them. The key features of the wave created from a sinusoidal function are:

• The Midline
  • The horizontal line which runs through the exact middle of the wave between the maximum points and minimum points.
• The Amplitude
  • The vertical distance from the midline to either the maximum or minimum points (which is the same value for both).
• The period
  • The distance on the X-axis between two consecutive maximum or minimum points.

The equation of a sinusoidal function is f(x) = (A)sin(P * x) + M where A equals the amplitude, P equals the period (calculated by dividing 2(pi) by P) and M equals the midpoint. The most difficult part to understand about this equation for me was how to find the period. I still don’t have a fully intuitive understanding of how it works but for the most part I’ve got it figured out. What helps me to find the period when doing these types of questions is to remember that when P > 1 the period becomes ‘shorter’ and when P < 1 the period becomes ‘longer’.

When working through sinusoidal graph questions, an important thing to remember is that an equation using Sin would ‘start’ at (0, 0) and curve up towards (1, 1) whereas a sinusoidal equation using Cos would ‘start’ at (1, 0) curve downwards towards (0, 1). It’s helpful to remember this in order to appropriately choose between Sin and Cos when looking at a graph to make which will make writing out the equation easier (either Sin or Cos can be used to describe any sinusoidal graph, however).

The rest of the unit had me finding the period, amplitude and midline of periodic graphs, writing out their full equations, creating a graph based on an equation, and finding a value of a given point in a sinusoidal function given it’s equation and/or graph. Some of the questions had me doing all of these steps at the same time. Going through these questions, I must have drawn ~40 graphs in my notebook which was very time consuming but really helped to understand how these equations worked. As I mentioned before, I had moments where I was incredibly frustrated but also had remarkable ‘break-through’ moments where things clicked and clarity set in. It wasn’t easy but it was very rewarding and satisfying once I figured out some of the more difficult concepts.

It’s looking unlikely that I’ll get through this course by the end of the month (i.e. by this coming Thursday). I still have two units remaining, Modelling (0/500 M.P.) and Rational Functions (0/1400 M.P.) and then will need to finish the course test which I think will be pretty tough in itself. I think partly why I got so frustrated this week was because it took me a long time to get through the unit and the longer it took the less likely I’d be to finish the course by the end of the month. It bothers me when I set a goal for myself and don’t achieve it. What helps me feel better is remembering that understanding the material is much more important than getting through it quickly. I’m also reminding myself that getting ~80-90% of the way towards a goal is still quite good and most likely much further than I would have got if I hadn’t set the goal in the first place!