Intuition For The Law Of Cosines

This is a companion discussion topic for the original entry at

well done kalid !"!!
please elobrote ,dot product is vector version of law of cosine …

now, if your general method really works, put up a proof of heron’s formula based on it.

Hi Susan, that’d be a great follow-up. I think this method probably works best for the Law of Cosines / Pythagorean theorem, but I’d like to see if it can simplify Heron’s formula (which is usually proved with a giant mess of algebra).

hi kalid,
it would be very useful.

I’m still processing this excellent tutorial. I almost got sidetracked–The square root of 900 is 30 :slight_smile:

Keep up the good work!!

@gulrez: Thanks! Hoping to do a follow-up on just that :).

@pat: Whoops, thanks for the typo! Just fixed.

I never post any comments but i had to do it now. You are awesome. These insights are so valuable. Good job.

@cjq: I make the diagrams in PowerPoint, hope that helps!

Hey, Kalid! I was just wondering how you make the diagrams you place in your lessons. You know, because it’s always better to place your thoughts into pictures than words, and I wanted to make my own!

@kalid the ideas that you have expressed in the post above (and in your geometric interpretation of complex numbers) are beautifully developed to their full potential in “Geometric Algebra”. Geometric algebra unifies geometry and algebra seamlessly and it encompasses complex algebra, quaternions, and many other seemingly disparate algebras. I have every hope that as your effort to elucidate and educate continues you will interestingly draw on the rich tradition of Geometric algebra. Look here for a very brief but accessible introduction “imaginary numbers are not real”

@susan Heron’s formula can be derived without resorting to the cosine law.

Let A be the area of the ∆ with sides a, b and c. Moreover, let h be the perpendicular height to the vertex from base b.

Begin with the well known
A = b*h/2

Now express h in terms of a,b and c.
This can be done by looking at the triangle in terms of the two right angled triangles partitioned by the altitude h.

b = √(a²-h²) + √(c²-h²)

Rearrange this equation to express h in terms of a,b and c and substitute back into the expression for the area. Heron’s result is obtained after identifying

s = (a+b+c)/2

and some algebra.

Hey, the article is simple and just too good. Thanks for this, I won’t be able to remember the formula anymore but everytime I will work out the interactions and understand what’s really going on.
Also looking forward for your follow-up on the difference in combinations of individual paths and whole areas.

ooooh my goodness that was amazing! Now I feel like everything in math must have some sort of logical explanation! It’s amazing that we assumed that the pythagorean theorem was some manifestation of the inherent magical properties of 90 degree angles in triangles, when it turns out that it’s everpresent in all triangles in a slightly different way.
Thank you so much Kalid! I loved this article, and I think it should be everyone’s introduction to the law of cosines!

Thanks Kenny, really glad to hear it clicked :).

I’m not a math person and want you to know how wonderful it is that you’ve put this information together online!

I will never understand how people just accept the language of math. It is arbitrary. Function? That is something attend in fancy dress. How does one approach a math question when there is no easy way to define an appropriate formula? Physics and geometry are understandable in real world terms. D=RT? I can illustrate that formula by throwing my algebra book in the trash. All the words in the formula are commonly used. I’ve been reading for hours and I still don’t see any meaningful definition of sine. I was excited to see your post about sine being a percentage, but I have to admit I began to lose the thread. A reader can usually find a relatable term to better understand an unfamiliar word in about two leaps. Ephemeral? Fleeting. Fleeting? Lasting only a short time. It seems like defining math terms takes one further away from any hope of solving a problem. Sine is a percentage of height, now I understand. Wait. You said there was a dome. Math is in space where there are negative numbers that are actually letters that are not part of any language that I speak or read. Is sine a percentage of my height? Is the floor level? I’m confusing myself. I think it sounded like perspective was involved. How far away was the caveman? Hold on. How is a percentage a curve? How do you just know that the numbers need to be squared? Why aren’t they cubed? Where is the word problem that helps me understand how finding the length of a two dimensional arc has anything to do with potential interactions? How can potential interactions be a static number? Wouldn’t potential outcomes lie on a bell curve? I’m so confused!

I found this site because I thought I might build a little bridge in my garden and wondered what it would look like if it were based on the golden ratio. The Internet informed me that there is such a thing as a golden angle; and now I’m stuck. I have no use for these lopsided triangles that illustrate cosine axr2 to the z or something. If I had a golden lamp I’d wish for an illustrated dictionary of math. In English. Where all the numbers are real and all the words are defined past the point of directing the reader to a similar, meaningless word. Cosine? Oh obviously it is the indirect inverse of a function of the degree of the sine. What was sine again? Sorry. Ugh! An hour after defining the terms of my problem I am no closer to a solution than I am to getting that golden lamp. Was this Heron’s problem too?

Q: Find the height/ Sagitta (bisect the symmetrical, obtuse triangle from the obtuse angle to the hypotenuse) using a “chord” length of nine feet and an obtuse angle of 137.508 degrees.

It very intuitive matter .Thanks . Now i am studying trigo and complex numbers ,its all right with the basics but when it comes to advanced concepts i am unable to see it intuitively . If you could suggest few books that take in very detail of the topics ,i would be obliged .