Cos (a-b) proof

Alexl

if anyone wants a nice way of proving Cos (a-b)

http://www.leavingcertsolutions.com/mall/leavingcertsolutions/pdf/lcm_p2_vector_proof_cos.pdf

Des23

That's amazing, normally that's an awful proof:D

ya-ba-da-ba-doo

Any chance you could do the proof as a comment? Im not able to go into the link? Thanks

Fince

theres a few handy bits and tips on that website, its a bit out of date though

MathsManiac

But there's a hell of a gap in that proof if you don't first prove the equivalence of the two expressions for the dot product.

Garf

Is that proof acceptable for full marks?

Alexl

ya-ba-da-ba-doo wrote: »

Any chance you could do the proof as a comment? Im not able to go into the link? Thanks

Its kind of hard beacuse theres a diagram

But its based on vectors. You set up your standard unit circle except with vectors. I.e. (Cos(a) i, Sin(a) j) (Cos(b) i, Sin (b) j)

Angle between two vectors is (A-B)
Then using dot product formula for A dot product b

Cos(A-B) = Cos(a)Cos(b) + Sin(a)Sin(b) / Modulus of a x Moduls of b

But modulus of a and and b= 1
Therefore
Cos(a-b) = Cos(a)Cos(b) + Sin(a)Sin(b)


Thats the best I can do really, I suggest you look it up.

Also a nice proof for Sin(A-B)

Set up unit circle with a(cosA sinA) and b(CosB SinB) as standard

Then use the two area of a triangle formulae.
1. Area =1/2 ( a x b x Sin C) where C= (A-B)
2. Area =1/2 (X1Y2 - X2Y1)
Set them equal to each other.
Hope these help

zodac

Are we allowed use this proof in the exam? It's seems awfully short

meathawk

I'm fairly sure you can't use vector methods to get the full marks.Well ,atleast that's what Johnny Brennan said.

MathsManiac

Garf wrote: »

Is that proof acceptable for full marks?

I seriously doubt it, unless you plug the big hole I mentioned.

ya-ba-da-ba-doo

Didnt want to start a new thread but does anyone know how to get two points on a quadrilateral when they give you two other points and the equation of one of the sides? Its really bugging me! (Its q 3 b(i) on the examcraft paper 2 btw)... thanks

MathsManiac

ya-ba-da-ba-doo wrote: »

Didnt want to start a new thread but...

Why?

ya-ba-da-ba-doo

MathsManiac wrote: »

Why?

Theres already too many and i thought seeing that people reading this one are probably pretty good at maths they'd be able to help? Soo anyone?

Alexl

zodac wrote: »

Are we allowed use this proof in the exam? It's seems awfully short

I really dont know whats wrong with it.
from what I've been told maths correctors are always happy to give full marks to new ideas. And you've fully proved it so why not??

About that hole maths maniac, every other proof on the course stems from another proof. Like the angle between two lines, you dont need to prove Tan(A+B), so why should this be any different

Alexl

meathawk wrote: »

I'm fairly sure you can't use vector methods to get the full marks.Well ,atleast that's what Johnny Brennan said.

Thats pretty funny seeing as I got the proof from his book

MathsManiac

Alexl wrote: »

I really dont know whats wrong with it.
from what I've been told maths correctors are always happy to give full marks to new ideas. And you've fully proved it so why not??

About that hole maths maniac, every other proof on the course stems from another proof. Like the angle between two lines, you dont need to prove Tan(A+B), so why should this be any different

Because the Tan(A+B) formula is one that you prove on the course, and clearly doesn't rely on the result you're trying to prove. In this case, you're using a vector result that might very well depend on the trigonometric one you're proving. So, it seems to me that you might need to establish that the vector result can be proven independently of the cos(A-B) formula, (which it can, by the way).

.:FuZion:.

ya-ba-da-ba-doo wrote: »

Didnt want to start a new thread but does anyone know how to get two points on a quadrilateral when they give you two other points and the equation of one of the sides? Its really bugging me! (Its q 3 b(i) on the examcraft paper 2 btw)... thanks

You pay attention to Perpendicular and Parallel slopes. I dont have my solution in front of me but thats what I did to get it. Also, the A Coordinates are ( 0, Y ) because its obviously on the axis.

MathsManiac, just while you're online I've a quick question for you.

I've covered the trigonometric proofs myself today (I wasn't in when we did them in school), and I just want to make sure it's ok to use the proofs perviously listed to prove the next one. For example, if the proof of Tan(2A) came up, is it safe to just sub A in for B in tan(A+B), or would you have to prove tan(A+B) first, if you get me.

I just want to make 100% sure, as I've only heard it from people who weren't exactly sure themselves. Thanks.

MathsManiac

-JammyDodger- wrote: »

MathsManiac, just while you're online I've a quick question for you.

I've covered the trigonometric proofs myself today (I wasn't in when we did them in school), and I just want to make sure it's ok to use the proofs perviously listed to prove the next one. For example, if the proof of Tan(2A) came up, is it safe to just sub A in for B in tan(A+B), or would you have to prove tan(A+B) first, if you get me.

I just want to make 100% sure, as I've only heard it from people who weren't exactly sure themselves. Thanks.

Yes, (you can use Tan(A+B) without proof). That was always my understanding when I was teaching, and was the case when I was an examiner on that paper many years ago. It also seems to be confirmed by the Chief Examiner in that letter to the support service about the log proofs of the calculus rules (referred to in another thread). When teasing out the acceptability of any proof in the contest of a syllabus, he says: "It also underlies the long-established practice that when proving trigonometric identities listed in the syllabus, candidates are entitled to use any of the identities that are listed before the one being proved, but not any of the ones after it."

So that looks like a solid endorsement of the practice.

That's perfect, thanks. I justed wanted to be 100% sure incase something along the lines of tan(2A) or cos(2A) came up (even though they're so simple they probably haven't ever came/wouldn't ever come up).

Thanks again.

Fringe

I found out about it in the reply about using the log rule to prove the product rule. If you can find the thread, it's in the file. Here's the exact quote:

"It also underlies the long-established practice that when proving trigonometric identities listed in the syllabus, candidates are entitled to use any of the identities that are listed before the one being proved, but not any of the ones after it."

EDIT: Damn!

MathsManiac

Fringe wrote: »

EDIT: Damn!

I hate da!

platypus

I'm pretty sure that vector proof IS acceptable for full marks