February 16th, 2019, 09:45 AM  #1 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7  Pythagorean theorem/Lesson Learned
Hello, I learned something interesting that I want to share. So, me and my brother were talking about math, and he told me that, "The Pythagorean Theorem only works for right triangles." I then thought to myself, "How would I find side length of a triangle like an isosceles from the two side lengths?" I then found this formula that answered the question: Side * 2cos(angle) However, I am the kind of guy that likes to find his own way. So, I had the idea. What if I modified the Pythagorean Theorem to fit other triangles types? Eventually, I got it. This is what I came up with: (a+b) sin(angle/2) I checked my work with a simple Isosceles Triangle. Side a = 1. Side b =1. The angles are 72, 54, and 54. According to the equation I said earlier, The answer should be 1.175570505. Then I tried the equation I made up. 1+1 is 2. The angle divided by 2 is 36. 2Sin(36) equals 1.175570505. It worked. I tried it with other side lengths, and it worked. Sides a and b = 2. Angles are 40, 40, and 100. Answer should be 3.064177772. The equation I made up. 2 + 2 is 4. 4sin(50) equals 3.064177772. It worked. Then, I tried something harder. A scalene with the following measurements. According to the law of sines, C=1, A=0.732050808, B=0.896575472. Angle AB=75. Angle BC=45. Angle CA=60 I tried to find Side C my own way. Here is what I came up with: a cos(angle AC) + b cos(angle BC) = c Checked my work. 0.732050808 cos(60)= 0.366025404 0.896575472 cos(45) =0.633974596. 0.366025404 + 0.633974596 = 1 It worked. I learned two things from this. First, I learned more about triangles and how to find out side lengths. Second, I learned there is more than one way to find an answer in mathematics. Some people use s*2cos(angle), I use my own way. Both ways work. I hope you enjoyed reading this, and if anyone has input, feel free to add it. Jared Last edited by skipjack; February 16th, 2019 at 05:47 PM. 
February 16th, 2019, 01:46 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,822 Thanks: 723 
The usual extension to the Pythagorean theorem is the law of cosines: $c^2=a^2+b^22ab\cos C$, where $C$ is the angle opposite side $c$.
Last edited by skipjack; February 16th, 2019 at 05:20 PM. 
February 16th, 2019, 03:13 PM  #3 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
Thanks for clarifying that. To be honest, I have difficulty understanding the law of cosines. I do use it, but sometimes I use it incorrectly. Mainly because it's hard for me to understand. Again, thanks for telling me that. I appreciate you and others here help me out with understanding mathematics. 
February 16th, 2019, 05:48 PM  #4  
Global Moderator Joined: Dec 2006 Posts: 20,969 Thanks: 2219  Quote:
formula becomes side * 2sin(angle/2). Quote:
That is usually written as c = a * cos(B) + b * cos(A). It's a standard result, but is often overlooked, perhaps because it doesn't seem to have a handy description or name.  
February 16th, 2019, 06:39 PM  #5 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
@skipjack, you are correct with everything you just said. Thank you for telling me that. You just made me realize something. I went and did days of research and experimenting to try to find my own way, and when I do find my own way, and I share it, it turns out it was already figured out by someone else. To be honest, I do not know how I am coming up with this stuff without even knowing that it already existed. I guess I just like experimenting and learning how things work my own way. I learn from experience. It's just who I am. Thanks for the help and the input. I appreciate it. Jared 
February 17th, 2019, 03:58 PM  #6  
Global Moderator Joined: May 2007 Posts: 6,822 Thanks: 723  Quote:
 
February 17th, 2019, 05:06 PM  #7 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
@mathman, I don't really understand it, because I didn't really learn about it well enough when I went through school. You have to understand I am mentally challenged, math is one of the area's I excel in, however, because of my disability, I was alternating between homeschool and regular school periodically. Because of this, I think I missed learning about certain math things, like the law of cosines. The Law of Sines, I just learned recently, to be honest. Would you mind explaining to me the law of cosines? It's difficult to understand because I never really learned it well enough. I know a little bit of it, but I don't know all of it. Jared 
February 17th, 2019, 05:59 PM  #8 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1038 
Google it; you'll get helpful sites like: https://betterexplained.com/articles/lawofcosines/ 
February 17th, 2019, 07:15 PM  #9 
Member Joined: May 2018 From: Idaho, USA Posts: 65 Thanks: 7 
Thanks, @Denis. I just looked it up. It's making more sense now. Thanks for helping out. Jared 
February 18th, 2019, 07:47 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 20,969 Thanks: 2219  cosine.jpg Code: c² = EB² + EA² = CB²  CE² + (CA  CE)² = a²  CE² + CA²  2CE*CA + CE² = a² + b²  2(a cos(C))*b = a² + b²  2ab cos(C) 

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