Maffs

[Xamonas Chegwé]

I was thinking about the cosine rule the other day and I came up with a proof based on rudimentary, vector mathematics. I have had a look around but I can't find anything exactly the same online - although it probably exists. Anyway, here it is...


Let ABC be a triangle, where a is the side opposite vertex A, b is opposite, B and c is opposite C.

Let a be the vector along side a from C to B.

Let b be the vector along side b from C to A.

Let c be the vector along side c from A to B.

Picture 3.png (3.94 KiB) Viewed 2606 times

Let the unit vectors i and j be parallel and perpendicular to a respecively.

So,

a = ai
and
b = b.cos(C)i + b.sin(C)j

So, by vector addition,

c = -b + a

= -(b.cos(C)i + b.sin(C)j) + ai

= (a - b cos(C))i - b.sin(C)j

Now, the magnitude (ie. length) of c can be determined from its resolved constituents using Pythagoras' Theorem

So:

c2 = (a - b cos(C))2 + (-b.sin(C))2

=> c2 = a2 - 2ab.cos(C) + b2.cos2(C) + b2.sin2(C)

Taking b2 out as a common factor leaves

=> c2 = a2 - 2ab.cos(C) + b2(cos2(C) + sin2(C))

and, using the identity sin2x + cos2x = 1 gives

=> c2 = a2 - 2ab.cos(C) + b2(1)

And, rearranging...

=> c2 = a2 + b2 - 2ab.cos(C)

Which just happens to be the Cosine Rule.

[JimC]

I love it!

I doubt whether it's the primary proof or derivation of the rule, more like a alternative pathway through the mathematical thickets...

[pErvinalia]

You lost me at "I was thinking"...

[Kristie]

You lost me at "I was thinking"...

[Xamonas Chegwé]

I love it!

I doubt whether it's the primary proof or derivation of the rule, more like a alternative pathway through the mathematical thickets...

Most of the proofs I have seen use similar steps but I have not seen one with exactly this path. I like that it combines basic vector manipulation with Pythagoras - I would be completely gobsmacked if it has not been proposed many times before but I think it is as neat as any other proof I have come across. I dreamt it up while waiting for a bus last week. It could make an interesting A level challenge question

...Hence, or otherwise, show that the cosine rule is true...

[JimC]

In our Specialist Maths, we do Trigonometry and Vectors as separate units (trig first, of course) but I keep trying to get students to see they are intimately connected...

[Xamonas Chegwé]

In our Specialist Maths, we do Trigonometry and Vectors as separate units (trig first, of course) but I keep trying to get students to see they are intimately connected...

My point.