Would you like to learn the Cosine Rule well?
Then keep watching.
G'day. Welcome to Crystal Clear Mathematics where it is easier than you think.
I'm your host, Graeme Henderson.
When we're learning to study triangles that do NOT contain a right angle,
there are three formulae that we need to know ...
three trigonometric formulae.
The first is the Sine Rule; second is the Cosine Rule;
and third is the Area Rule.
In this video, we're going to learn
how to memorise and learn and use the Cosine Rule effectively.
Now, interestingly, [for] all three rules we start with the same diagram ...
and here it is.
We draw a triangle that does not contain a right angle
and label its vertices A, B and C.
By convention, the side opposite A has length a,
the side opposite B has length b, the side opposite C has length c.
(remember, this is the entire length)
... and we draw an altitude (or height) and we call it h.
Now, unlike the previous two formulae ... I've already prepared those two videos ...
to derive the Cosine Rule, we need to split this length
(this base) into two parts.
We'll call this length x, which means that this length
(which makes up the whole length c) must be c-x.
That is, the two of them add up to c.
Now, the Cosine Rule, interestingly enough, is ... I should shade these differently ...
is really a trigonometric form of Pythagoras' Theorem.
And the reason I enjoy deriving this is because it drives into your mind
the understanding that Pythagoras' Theorem is the secret behind the Cosine Rule.
So, it's no secret that it comes out looking like Pythagoras' Theorem.
What is Pythagoras' Theorem for this right-angled triangle
on the left hand side?
Well, [it's] the square of the two short sides adds up to the square of the hypotenuse.
So, it would be h^2 + (c-x)^2 = b^2.
So, this squared plus this squared makes this squared.
What about this side ... this triangle?
Well, h^2 + x^2 must make this hypotenuse squared.
So, h^2 + x^2 = a^2.
So, we derive ... it's more than a derivation ...
we just write Pythagoras' Theorem for both the small right-angled triangles.
Now, straight away, you'll notice that both of these have an h^2 in them.
So, we can eliminate that, because we don't want h in our result.
So, what's h^2 equal to?
If I take this over here ... h^2 = b^2 - (c-x)^2.
If I do the same with this equation and get rid of the x^2,
I have h^2 = a^2 - x^2.
Now, if h^2 equals that and h^2 equals that,
and we've only got the one h in the equation ...
in the diagram ... then these must be equal.
So, b^2 - (c-x)^2 = a^2 - x^2.
Now let's expand both sides.
Well, we really only have to expand the left.
b^2 - ... I'll use parentheses ... c^2 - 2cx + x^2 = a^2 - x^2.
Removing the parentheses and noting that there's a minus sign in front
(it's very important to notice this) we get b^2 - c^2 + 2cx - x^2 = a^2 - x^2. 0:04:59.000,0:04:01.110 Now, this seems like a little bit of a jumble
but just remember that it was really just Pythagoras' Theorem and Pythagoras' Theorem
and remove the h^2.
We get to this and notice there's a -x^2 on both sides.
So we can remove them by adding x^2.
And I'm going to bring the c^2 over here.
So, I'm going to leave the b^2.
I'm going to have the a^2 here,
[bring the minus ...] add c^2 to both sides ...
and I'm going to subtract 2cx from both sides.
And that, now, is the form of the equation.
I've run out of [space] ... from the bottom of the board,
so I'm going to write it up here
on the [left] ... on the right hand side.
b^2 = a^2 + c^2 - 2cx.
Now, you can see that this looks like Pythagoras' Theorem, doesn't it?
Exactly!
But, it has another term to adjust for the fact
that we don't have a right-angle in this large triangle.
The problem is we have a formula with an x in it,
and we don't want to talk about a fraction or a portion of one side.
So, looking at this triangle on the right, what ratio can we use
to relate x to the side that we DO know ... that is, the complete length a?
Well, I think you can see that,
when we compare them with the angle here at B
we have the adjacent and hypotenuse ... and that's the cosine.
So, we can say, 'But CosB is adjacent over hypotenuse ... x/a.'
Which means, when I multiply both sides by a, that aCosB = x.
And that's what I substitute in here.
So, we had to go back to the right hand triangle
to get this trigonometric information to insert.
And now we can write this out as b^2 = a^2 + c^2 - 2 ...
now it's c times x and, since x is worth this,
c times this will be acCosB ... and there is our Cosine Rule.
Now, because of the constraints of the size of my whiteboard [and so on]
I haven't been able to derive it fully 'down the page.'
I could do it on paper, but I prefer standing here and showing you.
So, we've had to [sort of] do it in two columns [as it were].
This pattern is a very, very important one.
Notice that the side here has the same letter as the angle here.
So, in other words, if we're using angle B,
we can use it to find side b.
If this was CosA, this would have to be a^2.
If we were in some triangle where that was Cos Q,
this would have to be q^2. These must be the same letter.
Now, I hope you notice that these letters are the same here.
This pattern is universal ... [I will rub this out now].
So, if you had a triangle without a right-angle and ...
let's call the [corners] vertices Q, R and K ...
this side would be side k; this side would be side q;
this side would be side r.
And all that remains is for us to choose an angle and a side opposite.
Now if I wanted to relate it to K, this is the angle I'm using
and it means that these are the two sides I use.
And this is the pattern I look for.
I hope you can see it looks like a letter 'C' sitting on one of the corners.
'C' for cosine!
And all I would write ... I would start with k^2 ...
and there would be q^2 + r^2 - 2qr because they've got to be the same letters
... times CosK.
That's the way it's derived and written.
How would you learn it well?
You would learn it well by setting time aside without distractions
because you can focus and learn quickly that way.
Deriving this ... this one might take you two or three or four minutes ...
but deriving it perhaps once every afternoon for as long as it takes ...
it might only be two or three or four afternoons ...
it could take you a week and a half ... but, just every afternoon,
spend two to four minutes deriving it and then use it.
Use this to solve at least one or two questions
and, at the end of that time,
you should be able to ask yourself ... this is after you've set time aside
and after you've derived and used the formula ...
you ask yourself,
'Do I understand it well enough?' 'Am I fast enough?'
And, if you've done enough of this work
(what we call massed practice or intensive practice),
you'll say, 'Yes, I do understand, I can use it, and I am fast enough.'
You then ask the third question, 'How long before I'm likely to forget?'
... because forget you will.
Now, it might only take two or three weeks. It could take ten years, or thirty years ...
but, somewhere along the track, you'll forget because you haven't used it.
Now, most students, when they've first learned this,
will probably remember it for two or three weeks
or four to six weeks ... somewhere in there.
So, use your diary ... put a little entry in your diary
[to remember] to remind yourself to derive and use this formula again
in that number of weeks time.
And, if you use your diary to keep reminding yourself BEFORE you forget,
you will end up remembering it for a very, very long time indeed!
So, there it is ... Cosine Rule ... I hope it's helped,
and thank you very much for watching.
And, please, remember to subscribe!