In the “Measuring Your World” Project, we completed a series of worksheets that included finding the Sine, Cosine, Tangent, ArcSine, ArcCosine, ArcTangent, Law of Sines, and Law of Cosines. Some Worksheets were easier than others, but the solutions all included some form of the pythagorean theorem or the Law of Sines, Cosines, or Tangent.
The first thing that we did was prove the Pythagorean Theorem. We did this by putting squares on each side of of our right triangle, to make something that looks kind of like this:
The first thing that we did was prove the Pythagorean Theorem. We did this by putting squares on each side of of our right triangle, to make something that looks kind of like this:
Adding these squares gives us an idea of how large the sides are, and helps give an image of what the theorem states, which is that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs.
We Used the Pythagorean Theorem to derive the Distance Formula in one of our worksheets, and I didn’t fully understand it but after I got some help it made more sense. At first I just didn’t get it, but it turned out to be not that hard. If we had our triangle drawn out, we can use the pythagorean theorem to derive the distance formula fairly easily.
We used the distance formula a lot throughout this course, and we could also use it to derive the equation of a circle centered at the origin of a Cartesian coordinate plane. This made sense to me for the most part, but some parts didn’t come as easily. When we went further into detail with it, and started practicing it with some of our handouts, it started to make more sense.
The unit circle was something we used that help us find different angles and points. When looking at the circle, we figured out a rule that was “out = 90 degrees minus the in”, or minus theta. So we could figure out that the cosine of theta equals sin(90 degrees - theta), and this was also a trigonometric identity. The unit circle helped me a lot when I needed to look back at it for other problems, or just for reference.
We also had to find points on the unit circle at 30 degrees, 45 degrees, and 60 degrees. On the circle itself, there are lines at 30, 45, and 60 degrees that connected to points on the outside of the circle. For 30 it was (square root of 3/2, ½), for 45 it was (square root of 2/2, square root of 2/2), and for 60 it was (½, square root of 3/2). When we look at these we can see that 30 and 60 are the same just flipped. If we drew a triangle going through all three of these points, we could see that the angles recreate themselves along the x axis.
We also had to use the symmetry of the unit circle to find the remaining points. This wasn’t to difficult because we knew that at 30 degrees the points are (square root of 3/2, 1/2 ) and at 60 degrees the points were (½, square root of 3/2). If we know that these points are the same, just flipped, when the number is doubled (from 30 to 60), then it is fairly simple to find the missing points using the symmetry of the circle. Also, each quadrant of the circle has the same points, just flipped or changed somehow, but with the same numbers. If we have one quadrant filled out, then we know that the quadrant opposite of it will be the same but with negative or positive numbers, depending on which quadrant were looking at.
Another thing we used the unit circle for was defining sine and cosine. We could do this by drawing our triangles along certain points on the line within the circle, and finding the sins and cosines from there. If we are given certain angles, and know the lengths of certain sides than we can find both of these things. This was difficult for me at first but I eventually understood it and I think I am more comfortable doing it now. I needed some extra assistance when doing it but I feel like I have gotten better.
We also had to define the tangent function. We did this by dividing the opposite over adjacent. The tangent function is to find the hypotenuse or angle when you only have the opposite or adjacent side lengths, in which you can find the missing hypotenuse or angle. This was fairly simple for me to catch on with, and I think that it was a pretty simple concept once you understand it.
We can use similarity and proportion to find our cosine, sine, and tangent functions. If we look at the information that's given us (angles, lengths, etc.), we can determine our cosines, sines, and tangents through SOH-CAH-TOA. We can look at it as SOH, which is sine = opposite over hypotenuse. We can also look at CAH, which is cosine = adjacent over hypotenuse. Lastly theres TOA, which is tangent = opposite over adjacent. I think that I caught on to this pretty quickly, and understand it well.
We used the unit circle to define Arcsine, Arccosine, and Arctangent as well. That is when the -1 is in front of the sine, cosine, or tangent functions. It took me a while to understand the application of this, but once I did it got a lot easier to use. We’re able to use the unit circle to find it. During one of the SAT practice problems, I really understood this when I tried messing around with it, and ended up getting the right answer. That's when things really clicked with Arcsine, Arccosine, and Arctangent.
When we first got into Sine, cosine, and tangent, we did the Mount Everest problem. This problem really helped me further understand how these worked, and also helped me understand SOH-CAH-TOA further. We touched on this slightly last year as well, but I really didn’t get it until this problem. It really helped me understand it better, and showed me the correct way of doing it. I had an idea before, but I think my understanding of SOHCAHTOA, and sine, cosine, and tangent, has improved a lot this year.
We learned the Law of Sines a few weeks ago, and I think I am still learning it further. I think I have a grasp on it, but It could definitely be improved and I could definitely learn more about it. It was confusing at first because some aspects of it went straight in one ear and out the other. I feel like I have slight trouble understanding some of the content just because of how quickly we covered trigonometry, and how much information we receive in a short period of time. That's why I am glad I took notes in the process.
We also learned about the Law of Cosines, which I also slightly understood, but there were some parts that slipped away. I think if we had more time to work on these things, I would have had a better understanding of it all, but I am still happy with the knowledge I came away with in the end. I think learning the Law of Sines first helped make a path for the Law of Cosines because of how similar they are, and I think learning about the Law of Sines first helped me understand the Law of Cosines better. All in all I think I came out knowing a lot more about trigonometry since before I started. I barely knew a single thing about trigonometry before we started, so I can confidently say I learned a lot more about it in these past few weeks.
DP Update Part 2
In Dr. Drew's class, we all gave presentations to show the class what we measured, what techniques we used, and the final measurements that we got, along with some habits of a mathematician that we used.
Introduction:
My group and I chose to measure a snowman. This consisted of:
- 3 Spheres (Volume)
- 2 Eyes (Area)
- A Bowtie (Trigonometry)
- A Top Hat (Area)
- Carrot Nose (Trigonometry)
Math Calculations:
Area:
2 Eyes:
- Area (Circle) = (3.14) r squared
- 1 inch radius
- 3.14 x 1 = 3.14 in
Top Hat: - Area (Rectangle) = L x W
- 5 x 8 = 40
- Area (Circle) = (3.14) r squared
- A (big circle) = (3.14) x 8 = 25.12
- A (little circle) = (3.14) x 5 = 15.7
- 25.12 - 15.7 = 9.42 in
3 Spheres: Formula
12 inch Sphere: 904.8 In Cubed
24 inch Sphere: 7234.6 In Cubed
36 Inch Sphere: 24,416.64 In Cubed
Trigonometry
Bowtie:
h=√3/2⋅a
After substituting a=1 for 1 inch of the bowtie we have:
h=√3/2⋅1
h=√3/2
Carrot Nose:
Base: 2
Heigh: 5
side b: 5.0990195135928
base angle θ: 78.690
area S: 5
Reflection
How the work was divided:
Noah:
- Trigonometry
- Habits of a Mathematician: Collaborate and Listen
- Area
- Habits of a Mathematician: Take apart and put back together
- Volume
- Habits of a Mathematician: Conjecture and test
What would we do differently next time?
Probably pick a different object because this one was not as fun as we thought it would be to do. We might want to go for something on a larger scale like a building or something, or try doing something really small.