Definitive list of proofs for HL Maths PII

Jam-Fly

OK, here's what I have, please let me know what proofs are missing:


1. Equation of a tangent to a circle
The equation of the tangent to the circle x^2 + y^2 = r^2 at the point (x1, y1) on the circle is: x1.x + y1.y = r^2


2. Angle between two lines
If θ is the angle between two lines with slopes m1 and m2, then:
tanθ = (m1-m2)/(1+ m1.m2)

3. Cos (A+B)
Cos(A + = CosA.CosB - SinA.SinB

4. Perpendicular distance from a point to a line
The perpendicular distance, d, from the point (x1, y1) to the line ax + by + c = 0 is given by:
|ax1 + by1 + c| / sq.rt(a^2 + b^2)

5. Difference equations theorem
If α and β are the roots of the quadratic equation px^2 + qx + 2 = 0 and Un = lα^n + mβ^n for all n, then:
pun+2 + qun+1 + run = 0 for all n

6. Cosine Rule
In triangle abc, a^2 = b^2 + c^2 - 2bc.CosA


Then there are a load of possible Trig proofs. I THINK most of these can be answered just using and manipulating information from the log tables.

Please add any I've forgotten

aine-maire

Thank GOD the perpendicular distance proof came up last year!


I know that doesn't help formulate your list,but it does make it a little shorter;)

HQvhs

There are few semi-proofs in co-ordinate geometry. Proving a line segment is mapped onto another line segment and the like.

OxfordComma

Yeah, I agree about the line... There's 2 or 3 proofs involving transformations.

You have to be able to use the cos(A-B) formula to get sin(A+/-B) and tan(A+/B)... you also have to be able to get the formula for sin^2A and cos^2A as far as I know...


Do the Maclaurin series in further calculus count as proofs? Probably not.

awhir

anyone have a list of all the smaller proofs for example the tan(a+b)

Killaqueen!!!

The PIL formula, Tangent theorem AND cosine rule all came up last year.
So hopefully that will eliminate them from coming up this year. I bet they'll be mean and put one of the above on it again!

QueenOfLeon

Theres 4 line translations/proofs.

Theres a Line proof about 3 concurrent lines.

For some reason, i have the area proof (half abSinC) and the Sine formula (Sin A/a = Sin B/b) written out from ages ago....do we need to know these?

Its worth knowing that the (Cos squared A + Sine squared A = 1) formula uses the unit circle.

Tan (A+B) proof has a tricky bit in it that would probably be worth learning off.

These aren't proofs, but the (Cos(90 - A) = Sin A) things are handy.


Thats all i think!

Jam-Fly

out of interest, if you are prepared for nine questions, would it be possible to avoid doing any proof? I assume it wud be..

Fringe

Shame that a few proofs aren't coming up since a lot came up last year. There were barely on Paper 1 this year so I'm hoping some proofs come up. They're so easy once you know how to do it!

doorknob_opener

awhir wrote: »

anyone have a list of all the smaller proofs for example the tan(a+b)

Just listing them ...might take a while ...come back in 10 mins ...

professore

I think regurgitating proofs in an exam is a completely useless exercise from the point of view of testing mathematical ability. But that's the dumb system we are stuck with. Good luck with the exam.

Pygmalion

Jam-Fly wrote: »

out of interest, if you are prepared for nine questions, would it be possible to avoid doing any proof? I assume it wud be..

You probably could avoid them, but there's no guarantee that there'll be no proofs you'll have to do, just very, very unlikely.

If you really want to ace it learn the proofs that might come up on just one or two questions.

spoonbadger

If you think of the maclaurin series as proofs, then there's about 6 of those.

Even if you dont, it still doesn't hurt to learn the dividing idea in the tan-1 one.

doorknob_opener

awhir wrote: »

anyone have a list of all the smaller proofs for example the tan(a+b)

I tried making it look ''mathsy'', you know with fractions ect..., but the usual mathods I use on a different forum didn't work . Since I can't make powers, ^2 means to the powers of 2. And to make it clear, all angles are in brackets.

1. cos^2(A) + sin^(2) = 1

2. Cosine formula a^2 = b^2 + c^2 - 2bc cos(A)

3. cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

4. cos(2A) = cos^2(A) - sin^2(A)

5. sin(A+B) = sin(A)cos(B) + cos(A)sin(B)

6. sin(2A) = 2sin(A)cos(A)

This is where this might get hard to understand. By / I mean a fraction

7. tan(A+B) = [tan(A) + tan (B)]/[1 - tan^2(A)]

8. tan2A = [2tanA]/[1 - tan^2(A)]

9. cos2A = [1 - tan^2(A)]/[1 + tan^2(A)]

10. sin2A = [2tanA]/[1 + tan^2(A)]

11. cos^2(A) = 1/2[1 + cos(2A)]

12. sin^2(A) = 1/2[1 - cos(2A)]

If I typed anything wrong let me know. Good luck deciphering

ya-ba-da-ba-doo

Id say the proof for the angles between two lines might appear.. Or else the sin2A = 2tanA/1+tan^2A....

Zonda999

Lads, anyone know where i could find the maths papers from around 94 and 95 online??

Its just that apparently a few things on paper one came from the 94 paper so i thought they would definitely be worth a look!

yummy91

aine-maire wrote: »

Thank GOD the perpendicular distance proof came up last year!


I know that doesn't help formulate your list,but it does make it a little shorter;)

The fella on rte 306 said that it was answered terribly last year but, so they might put it on again,, they did stuff like that a few times now I think. It wont hurt to know it lik.

electriks

Anyone got tips for honours maths tomorrow for the leaving????

phasers

Study

electriks

Pygmalion

On a more serious note, I'd recommend you brush up on the option you're doing more than any other question, it's the one question you can't avoid or do somethign else instead of.

spoonbadger

Pygmalion wrote: »

On a more serious note, I'd recommend you brush up on the option you're doing more than any other question, it's the one question you can't avoid or do somethign else instead of.

+1 on this. And to be honest, you have enough time tonight to go over the whole paper before tomorow.

Pick a hard exam year and do out the paper. Dont bother trying to mark it, just make a note of any questions you had trouble with and do them out of your notes. Best way to study maths, regardless of how much time you have! .

Most importantly, asleep before 11:30.

In terms of what to study in particular...

Trigonometry and line proofs
Trig identities
Maclaurin series
Integration by parts
Max/Min questions
Vectors
Solving 3d triangles.

After that, just practice some probability. If you can do all that, it'll at least pull your ass above the pass line.

yummy91

Dont panic,,,, being positive really does help.
Know your probability well and your option, go over as many examples of them as you can and go over the proofs.
I think it helps to go over the things you're worst at in the morning before you go in, like get up about a hour early,, but it depends on what works for you.
Good luck anyway:)

AND look in the inspiration thread, :):).

ya-ba-da-ba-doo

Here how do u get this? Any kind of formula or is it simply just your ability to see the pattern? Hope it doesnt come up tomorrow! HELP PLEASE!

robz18

ya-ba-da-ba-doo wrote: »

Here how do u get this? Any kind of formula or is it simply just your ability to see the pattern? Hope it doesnt come up tomorrow! HELP PLEASE!

You need to be able to see the pattern. It's not tough at all. Look at a few questions and answers, you'll get the hang of it in no time.


Remember that if the signs change the general term will need to be multiplied by a -1 to the power of something.
The x is usually to the power of 2n plus or minus something too.

Just remember to always check it for every term (well, not every term obviously, but at least 4 or 5) in the series before you move on to the next question.

ya-ba-da-ba-doo

^cheers

LivingDeadGirl

FFS my teacher never taught us any of these!!

galway.gaa

what order are ye doin the questions in? i was thinkin 8 6 7 1 2 3

robz18

I'm going to do 2, 8, 4, 5, 3 and then 1.

Killaqueen!!!

what are all these tan proofs that people keep talkin about? they're not in my book and can't find it in my notes..don't remember doing that!

Des23

Has the difference equation proof ever come up?
Could definitely see it happening, people usually take it as a straightforward 20 marks, if they are out to mix things up a bit...

Prowetod

The order I will do the questions will most likely be: 8,1,4,5,6,2,3.

Does anyone know if the line transformation proofs come up in Q3 often?

awhir

doorknob_opener wrote: »

I tried making it look ''mathsy'', you know with fractions ect..., but the usual mathods I use on a different forum didn't work . Since I can't make powers, ^2 means to the powers of 2. And to make it clear, all angles are in brackets.

1. cos^2(A) + sin^(2) = 1

2. Cosine formula a^2 = b^2 + c^2 - 2bc cos(A)

3. cos(A+B) = cos(A)cos(B) - sin(A)sin(B)

4. cos(2A) = cos^2(A) - sin^2(A)

5. sin(A+B) = sin(A)cos(B) + cos(A)sin(B)

6. sin(2A) = 2sin(A)cos(A)

This is where this might get hard to understand. By / I mean a fraction

7. tan(A+B) = [tan(A) + tan (B)]/[1 - tan^2(A)]

8. tan2A = [2tanA]/[1 - tan^2(A)]

9. cos2A = [1 - tan^2(A)]/[1 + tan^2(A)]

10. sin2A = [2tanA]/[1 + tan^2(A)]

11. cos^2(A) = 1/2[1 + cos(2A)]

12. sin^2(A) = 1/2[1 - cos(2A)]

If I typed anything wrong let me know. Good luck deciphering

do you have to learn all these or could u just prove them using common sence ?

gota study all of Q8,Q4,Q5 in 3hours its gona be tough !

Killaqueen!!!

Des23 wrote: »

Has the difference equation proof ever come up?
Could definitely see it happening, people usually take it as a straightforward 20 marks, if they are out to mix things up a bit...

2007

It's an annoying enough proof...what it is your actually proving? I know how to do it but I dunno what it is I am doing! It ends up equal to zero or something..

sock puppet

Not really sure how I'm going to go in this paper. I always go way over time on the circle question. I'll probably make it up on probability though.

And can someone help me with the proof of sin(a+b)? My book says to use the formula for cos(a-b) and substitute (pi/2 - a) in for a. Does cos(pi/2-a) = sin a?

platypus

sock puppet wrote: »

Not really sure how I'm going to go in this paper. I always go way over time on the circle question. I'll probably make it up on probability though.

And can someone help me with the proof of sin(a+b)? My book says to use the formula for cos(a-b) and substitute (pi/2 - a) in for a. Does cos(pi/2-a) = sin a?

Yes for this proof you need to remember cos(90-A)=sinA and sin(90-A)=cosA

sock puppet

sock puppet wrote: »

Not really sure how I'm going to go in this paper. I always go way over time on the circle question. I'll probably make it up on probability though.

And can someone help me with the proof of sin(a+b)? My book says to use the formula for cos(a-b) and substitute (pi/2 - a) in for a. Does cos(pi/2-a) = sin a?

Sin(90-A) is equal to cos(A), likewise, cos(90-A) = sin(A). It makes sense if you do a few examples on your calculator. For example:

Sin(90-30) = cos(30); sin(60) = cos(30). It's basically because, simply, cosine and sine are two sides of the one coin.

ya-ba-da-ba-doo

^ yeah cos(pi/2-A)=sinA and sin(pi/2-A)=cosA

xOxSinéadxOx

awhir wrote: »

do you have to learn all these or could u just prove them using common sence ?

gota study all of Q8,Q4,Q5 in 3hours its gona be tough !

no you don't have to learn them. they're just the trig identities
the proofs are
Circle:
Tangent Theorem

Line:
Angle between 2 lines
Perpindicular Distacen

Trig:
Cos^2A + Sin^2A = 1
compund angle formulae
cosine rule

Discrete Maths:
difference euqation theorem

trying to learn these all now :(

spent all day yesterday studying maths and I'm still screwed. I've done no irish either
aarrrgh