In trigonometry, students are constantly working with finding missing side lengths and angles. When in high school, my trigonometry course talked highly of the Pythagorean Theorem, but did not put as much emphasize on the law of sine and cosine. However, I think the laws are just as important, which is why I choose to discuss the laws of sine and cosine for this upcoming project.
As of now, I understand that the laws of sine and cosine do not work in relation to right triangles. Therefore, the laws will not be accurate if a 90-degree angle exists within the triangle. They function from looking at angles and matching to the line across, labeling it as its correspondent. I know the formula as (a/Sin A) = (b/Sin B) = (c/Sin C). However, it can also be written vice versa ie. (Sin A/a) = (Sin B/b) = (Sin C/c). With that said, this is how the formula is written in textbooks, but the entirety of the formula is not used at a time. For example, if angle A, side length a, and angle B is given, then we do not use (Sin C/c) or (c/Sin C) because it is irrelevant at this time when finding side length b. In terms of the law of cosine, I can comprehend the formula which is shown below. The missing side length squared equals the sum of both the known side lengths squared minus two times the two known side lengths times cosine of the unknown side lengths matching angle. I also know that even though the side length is what the formula is stating each formula is missing, it can still be computed to find a missing angle measure. It just requires more math thought processes to occur.
In contrast, there are aspects that I wish to learn about the laws of sine and cosine. I would like to know the history such as who discovered the laws and how did they come to understand that the laws worked. I think it is very important for students to understand where mathematical concepts come from, and not just how to compute using the formula. In terms of the law of sine, I would also like to engage in the idea of doing cross multiplication first and then plugging in numbers. I wonder if the same solutions would be computed. I think they would, but it is an interesting way to use the laws of sine that intrigued me. I would also like to engage in proofs of how the laws of sine and cosine came to be.
Interesting Resources:
Hi Miranda!
ReplyDeleteI will give you credit on doing this topic. I know personally for me in high school this was the class that gave me the most trouble and this initially was a more challenging topic for me. Anyways, it seems to me you have some very good resources already on this topic and I really like your plan. I specifically like that you are incorporating the history of these formulas to go along with how to use them, as this may have been more useful and kept me engaged with the topic when I was in school. I like the plan and am very curious to see where your project will lead you as we continue through this class.
-Austin
Hi Miranda,
ReplyDeleteGreat topic choice! Pythagorean's Theorem has so many intricate piece within it and related to it, like you mentioned the law of sines for example. A few other break off topics/ ideas could be inverse sine = cosecant and SOH COA TOA. I definitely recommend discussing the visual proof of Pythagoreans Theorem that we discussed in Suzanne's course. I linked a website below that goes through the details.
http://www.cut-the-knot.org/pythagoras/
Hello Miranda,
ReplyDeleteI think it's cool that you're doing this as your topic! I definitely did better in science and social studies than I ever did in math, but I think you seem very prepared to dive in deep into this topic! Some of the questions you had at the end seemed really interesting, regarding the history of the pythagorean theorum. It would be kind of cool to see how you could use visual aids to help students comprehend what's going on with this lesson!
Like the others stated above, I too am looking forward to seeing where you take this topic. I recall learning trigonometry myself in high school yet not liking it much until a friend started to explain its connections to geometry. I loved geometry and found that was my entry point into it.
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