It felt like I was having not a very productive week as I was working through KA this week but in hindsight I actually got quite a bit done. I spent the first few days reviewing trig identities – specifically the angle addition identities – which is what made me feel like I wasn’t making much progress through Parametric Equations, Polar Coordinates, and Vector-Valued Functions. By the end of the week, I ended up getting through 5 videos and 3 exercises which could have been more but which I’m happy with, nonetheless.

In some of the exercises I worked through last week (i.e. Week 162), a few of the answers from KA used the sine and cosine double angle identities, which themselves are an extension of the angle addition identities. I couldn’t remember why they worked and I didn’t want to just blindly use the formulas so I spent all of Tuesday and some of Wednesday reviewing the proofs for these identities. By the end of Wednesday I was able to get a much clearer picture of why they work. I can now visualize what’s going on with these identities and have a much stronger grasp on WHY they work. Here are two very messy pages from my notes of the proof for the sine and cosine angle addition identities:

I continued on to review a bunch of other trig identities on Wednesday which all seemed much more straightforward. It seems that the main trig identities are: 

• a² + b² = c²
• sin(a + b) = sin(a)cos(b) + sin(b)cos(a), and
• cos(a + b) = cos(a)cos(b) – sin(a)sin(b)

Using these three fundamental identities, you then use some algebra to get this list of identities:

After reviewing those identities, on Wednesday I was also able to get through the two videos in the section titled Area: Polar Regions (Single Curve). Using integrals with polar coordinates has been tough to wrap my head around. One thing I did that helped was Google “why are they called polar coordinates?”. The first result said something along the lines of, imagine that there’s a pole at the origin of a grid and a rope is being pulled around it in a circle. Thinking about it like this, the polar coordinate grid on Desmos started to make a lot more sense:

The videos from this section taught me the formula used to find the area between a function and the origin using polar coordinates. The main difference when finding area with polar coordinates is that the area is part of a circle, i.e. it looks like a slice of a pie. Instead of using the formula for the area of a circle, πr², you multiply that formula by the ratio of the angle being measured over the perimeter of a circle, a.k.a. θ/2π, which ends up being θ/2π * πr² = θr²/2. The actual formula used to calculate the area using polar coordinates is _a∫^b (r(θ))²/2 dθ and I’m still a bit confused as to why the integrand isn’t θr²/2. I’m pretty sure it’s because dθ takes the place of θ which essentially says you’re taking all the tiny, tiny changes in θ between a and b and summing them up.

I used this formula in the exercises from the following section, Area: Polar Regions (Two Curves), as well. Here are three example questions I worked on between the 3 exercises from those two sections:

Question 1

Question 2

Question 3

I started the following section, Arc Length: Polar Curves, and made it through the two videos from that section. I still have to do the exercise in it so I’m not going to bother getting into what I learned from those two videos until next week after I’ve worked through the exercise and have a better understanding of how it all works.

I have exactly 15 days left in October and have one video and two exercises left to get through in Parametric Equations, Polar Coordinates, and Vector-Valued Functions (1,040/1,500 M.P.). I think I should be able to get through the video and both exercises early this coming week and so I’ll stand a pretty good chance of finishing off this unit before the end of the month. It will depend on how many attempts it takes me to get through the unit test, but I feel pretty good about everything I’ve learned in this unit so I’m hoping I won’t find it too difficult. I’m assuming if I do get through this unit before the end of the month it won’t be until the following week, but who knows! Maybe I’ll be starting off next week’s blog by saying I’m on the FINAL. UNIT. OF CALCULUS. TWO!!! 😱 😱 😱

Probably not though.