Trigonometry, calculus proofs?

Leaving Cert Student

Do we need to know all the first principles proofs, the product rule, quotient rule proofs etc. and all the trig proofs too? I know for geom. we only need to learn 3 of them so is there something similar for trig and calculus or? Thanks to anyone who replies

Monsieur Folie

Yes the calculus proofs and trig proofs can all be asked for.

Also, in geometry, you can be asked for a number of the theorems you've done in Junior Cert so it's not just those three. Off the top of my head I think they are: Three angle's in a triangle add to give 180, the exterior angle of a triangle is equal to the two interior opposites, opposite sides and angles in a parallelogram are equal, Pythagoras' theorem, and the angle at the centre of a circle is twice that of the angle on the circle when they're standing on the same arc.

They're all easy but my teacher has said they can be asked and they are all in the appendix of my maths book so I've no reason to doubt her.

Leaving Cert Student

Monsieur Folie wrote: »

Yes the calculus proofs and trig proofs can all be asked for.

Also, in geometry, you can be asked for a number of the theorems you've done in Junior Cert so it's not just those three. Off the top of my head I think they are: Three angle's in a triangle add to give 180, the exterior angle of a triangle is equal to the two interior opposites, opposite sides and angles in a parallelogram are equal, Pythagoras' theorem, and the angle at the centre of a circle is twice that of the angle on the circle when they're standing on the same arc.

They're all easy but my teacher has said they can be asked and they are all in the appendix of my maths book so I've no reason to doubt her.

Thanks dude, had heard it was only 3 of them that could be asked for geom oh well...
Some of those calculus ones are so hard to remember too I dont mind the trig ones though!

Slow Show

Monsieur Folie wrote: »

Also, in geometry, you can be asked for a number of the theorems you've done in Junior Cert so it's not just those three. Off the top of my head I think they are: Three angle's in a triangle add to give 180, the exterior angle of a triangle is equal to the two interior opposites, opposite sides and angles in a parallelogram are equal, Pythagoras' theorem, and the angle at the centre of a circle is twice that of the angle on the circle when they're standing on the same arc.

Is it not just that you need to know how to apply them, not to formally prove them?

Leaving Cert Student

Slow Show wrote: »

Is it not just that you need to know how to apply them, not to formally prove them?

That's what I had hoped and just to be able to prove 3 of them?

MathsManiac

Syllabus says:

Students will be expected to understand the meaning of the following terms
related to logic and deductive reasoning: Theorem, proof, axiom, corol-
lary, converse, implies, is equivalent to, if and only if, proof by
contradiction.

A knowledge of the Axioms, concepts, Theorems and Corollaries pre-
scribed for JC-HL will be assumed.

Students will study all the theorems and corollaries prescribed for LC-OL,
but will not, in general, be asked to reproduce their proofs in examination.
However, they may be asked to give proofs of the Theorems 11, 12, 13,
concerning ratios, which lay the proper foundation for the proof of Pythago-
ras studied at JC, and for trigonometry.

They will be asked to solve geometrical problems (so-called \cuts") and
write reasoned accounts of the solutions. These problems will be such that
they can be attacked using the given theory. The study of the propositions
may be a useful way to prepare for such examination questions.

That suggests, I think, that only those three proofs would be asked.

Monsieur Folie

Thanks for the clarification there. So we only need to know the formal proofs for the last three? I stand corrected then. :P

Still can't hurt to know the others though.

skippy1977

I've just finished a set of notes for the students in my class and I've compiled the following list.....which I stand by with absolutely no degree of certainty. The syllabus is a nightmare to try and extrapolate information from and the Project Maths website is a maze of random resources.

I've given mine:

De Moivre's Theorem
Differentiation by 1st Principles (6 of them)
Other Differentiation Proofs (4 of them General, Sum, Product, Quotient)
A Proof by Contradiction (Root 2)
Trigonometric Identities (8 of them 1-7 and 9)
Theorems 11,12,13 definitely (and 4,6,9,14,19 just in case?!?)

They have a list of definitions such as:
if and only if
converse of a theorem
is equivalent to
etc

There are 3 types of Proof by Induction you also need to know how to do. Series, Division and Inequality though these can't be learned off by heart as you can be asked various (infinite) versions of each.

I'm also getting mine to learn off:

Derivation of Cone and Sphere formula using Integration
and De Moivre's Theorem to prove a Trigonometric Identity
and a few other small bits that aren't really learned proofs but can be difficult to come up with yourself on the day of the exam.

I'm convinced I've left something out but that's the situation we've been left with. Hope it helps a little...

Oh and there are a couple of posts with a similar heading so I hope people don't mind that I've duplicated the post in a few places. Although more than one duplicate would be a triplicate?? Replicate....where are the English teachers

James

Monsieur Folie

That's a very helpful post.. Your list pretty much overlaps with all the stuff I have done with my teacher, but it's nice to see it all together like that. Might go and make a check list now. :P

Leaving Cert Student

skippy1977 wrote: »

I've just finished a set of notes for the students in my class and I've compiled the following list.....which I stand by with absolutely no degree of certainty. The syllabus is a nightmare to try and extrapolate information from and the Project Maths website is a maze of random resources.

I've given mine:

De Moivre's Theorem
Differentiation by 1st Principles (6 of them)
Other Differentiation Proofs (4 of them General, Sum, Product, Quotient)
A Proof by Contradiction (Root 2)
Trigonometric Identities (8 of them 1-7 and 9)
Theorems 11,12,13 definitely (and 4,6,9,14,19 just in case?!?)

They have a list of definitions such as:
if and only if
converse of a theorem
is equivalent to
etc

There are 3 types of Proof by Induction you also need to know how to do. Series, Division and Inequality though these can't be learned off by heart as you can be asked various (infinite) versions of each.

I'm also getting mine to learn off:

Derivation of Cone and Sphere formula using Integration
and De Moivre's Theorem to prove a Trigonometric Identity
and a few other small bits that aren't really learned proofs but can be difficult to come up with yourself on the day of the exam.

I'm convinced I've left something out but that's the situation we've been left with. Hope it helps a little...

Oh and there are a couple of posts with a similar heading so I hope people don't mind that I've duplicated the post in a few places. Although more than one duplicate would be a triplicate?? Replicate....where are the English teachers

James

Looks pretty good to me, thank you very much for a list, feels good to finally see one... Some of the differentiation proofs are awful though and I haven't seen them in any of the sample papers really? The general, sum and product ones.
The proof by contradiction we haven't done at all yet? Little worried about that..

I'd be interested to know the little extra bits you get your class to learn off as all I can think of is the de moivres proofs they can ask for...?

skippy1977

Yeah hopefully they will steer away from asking those differentiation proofs especially as they are from the old course. The following application of De Moivre was very popular on the old course and I've seen it on recent sample papers. Would be very hard to come up with on the day.


As would this special case in Integration

Leaving Cert Student

skippy1977 wrote: »

Yeah hopefully they will steer away from asking those differentiation proofs especially as they are from the old course. The following application of De Moivre was very popular on the old course and I've seen it on recent sample papers. Would be very hard to come up with on the day.


As would this special case in Integration

I might be a bit strange but I kinda like those de moivre proofs...
As for the integration, I hate that special case... just wondering when you set x = 3sinQ, did you choose the coefficient to be 3 just because it is the square root of the other number under the root, 9?

skippy1977

Yep if the number under the root was 16 you'd pick 4.