Theorems/proofs for Monday

Daniel S · 11-06-2011 04:40PM #1

What theorms/proofs do we need to know for Monday?

Am I missing any (I'll update this post if we find anymore):

Also, if there's any there that won't come up, I'd apprieciate it if you'd comment below.

Equation of a tangent to a circle.

Angle between to lines.

Perpendicular distance

Difference equation

1. Cos2A + Sin2A = 1
2. Cosine formula: a2 = b2 + c2 ⎯ 2 bc cos A
3. Cos (A+B) = cos A cos B ⎯ sin A sin B
4. Cos 2A = cos2A ⎯ sin2A
5. Sin (A+B) = sin A cos B + cos A sin B
6. Sin 2A = 2 sin A cos A
7. Tan (A+B) = tanA + tanB / 1-tanAtanB
8. Tan 2A = 2tanA/1-tan²A
9. Cos 2A = 1-tan²A/1+tan²A
10. Sin 2A = 2tanA/1+tan²A
11. Cos2A = (1 + cos 2A)/2
12. Sin2A = (1 – cos 2A)/2
13. 2 cos A cos B = cos (A+B) + cos (A⎯B)
14. 2 sin A cos B = sin (A+B) + sin (A⎯B)
15. 2 sin A sin B = cos (A⎯B) – cos (A+B)
16. 2 cos A sin B = sin (A+B) – sin (A⎯B)
17. Cos A + cos B = 2cos(A+B)/2cos(A-B)/2
18. Cos A – cos B =-2sin(A+B)/2sin(A-B)/2
19. Sin A + sin B =2sin(A+B)/2cos(A-B)/2
20. Sin A – sin B =2cos(A+B)/2sin(A-B)/2

polka dot · 11-06-2011 04:52PM

Difference equation proof

Daniel S · 11-06-2011 04:54PM

polka dot wrote: »

Difference equation proof

Have you got a link to it? I can't find it.

polka dot · 11-06-2011 04:58PM

mtb_kng wrote: »

Have you got a link to it? I can't find it.

Um... not yet. It came up in 2007.

Daniel S · 11-06-2011 05:00PM

polka dot wrote: »

Um... not yet. It came up in 2007.

polka dot · 11-06-2011 05:07PM

If a and b are the roots of the quadratic equation:
px^2 + qx + r = 0
and Sn = la^n + mb^n, for all n,
then pSn+2 + qSn+1 + rSn = 0 for all n.

(those ns should be the little ones below the letter...)

Proof:

Because a is a root of px^2 + qx + r = 0,
pa^2 + qa + r = 0
Likewise: pb^2 + qb + r = 0
Then Sn+1 = la^(n+1) + mb^(n+1)
and Sn+2 = la^(n+2) + mb^(n+2).

Thus: pSn+2 + qSn+1 + rSn

= p [la^(n+2) + mb^(n+2)] + q[la^(n+1) +mb^(n+1)] + r[la^n + mb^n]
= l [pa^(n+2) + qa^(n+1) + ra^n] + m[pb^(n+2) + qb^(n+1) + rb^n]
= la^n [ pa^2 + qa + r] + mb^n[pb^2 + qb + r]
= la^n (0) + mb^n (0)
= 0

*

There you go.

Edit: Snap :P Didn't see someone had posted it before I posted. I'll leave this here anyway though.

biggaman · 11-06-2011 06:57PM

Is that them all, then?

Daniel S · 11-06-2011 07:43PM

biggaman wrote: »

Is that them all, then?

I don't know, maybe. I want to know myself.

Daniel S · 11-06-2011 07:47PM

Michael_E wrote: »

Nope - book I'll list them now for ya:
Perpendicular Distance - Text and Tests 4, pages 93-95.

Slope between two lines (tan A =....) - Text and Tests 4, pages 98-99

Equation of a tangent to a circle - Text and Tests 4, page 356

cos^2x+sin^2x=1 - This I didn't use the book, I worked it out myself. I'll be glad to do it out and show you if you like

Cosine Rule - Got a nice proof online, here. I thought that the triangle on the left is easier, I thought. [.pdf format]

cos (A+B) and related proofs - Text and Tests 4, pages 144-146

cos 2A = cos^2A-sin^2A - Did this myself. Just did cos(A+A) and worked from there.

Sin2A - Same story as cos2A, above.

tan2A=2tanA/1-tan^A - Same, tan (A+A). On page 150 of Text and Tests 4.

cos2A (and sin2A) into tan - Text and Tests 4, page 152-153

cos^2A (and sin^2A) into 1/2(1 +/- cos2A) - Text and Tests , page 150.

Difference equations - came up last year, but it's in Text and Tests 5, page 189.

Linear transformation proofs (that f maps line to a line, and a pair of || lines to another pair of || lines) - Text and Tests 5, pages 201-202

.

Prinkle · 11-06-2011 08:08PM

For the equation of a tangent to a circle, that's a circle with centre (0,0), right?

cocopopsxx · 11-06-2011 08:54PM

Prinkle wrote: »

For the equation of a tangent to a circle, that's a circle with centre (0,0), right?

Ya, according to my notes anyway!

Kedo93 · 11-06-2011 09:27PM

mtb_kng wrote: »

Differential equations - came up last year, but it's in Text and Tests 5, page 189.

Ehm I presume you mean difference equations yeah? Differential equations aren't on the syllabus right?

River Song · 11-06-2011 09:31PM

Prinkle wrote: »

For the equation of a tangent to a circle, that's a circle with centre (0,0), right?

Yep indeedy. x^2+y^2=r^2

Kedo93 wrote: »

Ehm I presume you mean difference equations yeah? Differential equations aren't on the syllabus right?

Yes, my apologies. I am forever getting the two things mixed up. I'll fix the original post right now.

GV_NRG · 12-06-2011 12:21AM

mtb_kng wrote: »

Equation of a tangent to a circle.

Angle between to lines.

Perpendicular distance

for these must we just know the formula for them??

mtb_kng wrote: »

1. Cos2A + Sin2A = 1
2. Cosine formula: a2 = b2 + c2 ⎯ 2 bc cos A
3. Cos (A+B) = cos A cos B ⎯ sin A sin B
4. Cos 2A = cos2A ⎯ sin2A
5. Sin (A+B) = sin A cos B + cos A sin B
6. Sin 2A = 2 sin A cos A
7. Tan (A+B) = tanA + tanB / 1-tanAtanB
8. Tan 2A = 2tanA/1-tan²A
9. Cos 2A = 1-tan²A/1+tan²A
10. Sin 2A = 2tanA/1+tan²A
11. Cos2A = (1 + cos 2A)/2
12. Sin2A = (1 – cos 2A)/2
13. 2 cos A cos B = cos (A+B) + cos (A⎯B)
14. 2 sin A cos B = sin (A+B) + sin (A⎯B)
15. 2 sin A sin B = cos (A⎯B) – cos (A+B)
16. 2 cos A sin B = sin (A+B) – sin (A⎯B)
17. Cos A + cos B = 2cos(A+B)/2cos(A-B)/2
18. Cos A – cos B =-2sin(A+B)/2sin(A-B)/2
19. Sin A + sin B =2sin(A+B)/2cos(A-B)/2
20. Sin A – sin B =2cos(A+B)/2sin(A-B)/2

and for these must we just know that they exist or what??

sorry if its a stupid question but im really confused

B_Fanatic · 12-06-2011 12:25AM

Are some of these identities? Not 100% on the identity/proof thing.

Daniel S · 12-06-2011 09:11AM

GV_NRG wrote: »

for these must we just know the formula for them??

and for these must we just know that they exist or what??

sorry if its a stupid question but im really confused

You have to be able to prove those identities and formulas.

spurious · 12-06-2011 01:50PM

mtb_kng wrote: »

What theorms/proofs do we need to know for Monday?

They are all listed on the syllabus.

Shanee. · 12-06-2011 02:09PM

Difference equations are on the course?

snicket · 12-06-2011 02:12PM

Does anyone know the proof for Cos (A+B) = cos A cos B ⎯ sin A sin B? My teacher explained it one way using Cos(A-B) = Cos AcosB + sinAsinB and then just change the sign of B, but in my revision book (which could be out of date) there's a proof using the unit circle. Which one is better?

hunii07 · 12-06-2011 02:14PM

snicket wrote: »

Does anyone know the proof for Cos (A+B) = cos A cos B ⎯ sin A sin B? My teacher explained it one way using Cos(A-B) = Cos AcosB + sinAsinB and then just change the sign of B, but in my revision book (which could be out of date) there's a proof using the unit circle. Which one is better?

Do whichever one you're more familiar or comfortable with... my text book has the one using the unit circle and I find that one good ...but it's up to you..:)

snicket · 12-06-2011 02:18PM

Thanks

The one my teacher gave us seemed way too simple to be a proper proof though... might go over the other one anyway just in case!

hunii07 · 12-06-2011 02:20PM

snicket wrote: »

Thanks The one my teacher gave us seemed way too simple to be a proper proof though... might go over the other one anyway just in case!

It's not that bad anyway ..looking at it first is a bit daunting but over all it's pretty straight forward

polka dot · 12-06-2011 02:26PM

Ok I was doing some papers and this question came up in 2006. Question 7 (c)

The mean of the integers from −n to n, inclusive, is 0.
Show that the standard deviation is

sq. rt n(n+1) / 3

Is this a proof, theorem, something that may come up?

partyatmygaff · 12-06-2011 02:40PM

Nope, that question is just an example of what an abstract statistics question may be like. If you need a hint, it involves the natural number series from Q4 Paper I.

rosskind · 12-06-2011 02:41PM

If you are asked to prove something like tan(A+B), you can assume cos(A+B)=cosAcosB-sinAsinB, yeh?[FONT=Times New Roman,Times New Roman][FONT=Times New Roman,Times New Roman]
[/FONT][/FONT]

hunii07 · 12-06-2011 02:50PM

rosskind wrote: »

If you are asked to prove something like tan(A+B), you can assume cos(A+B)=cosAcosB-sinAsinB, yeh?[FONT=Times New Roman,Times New Roman]
[/FONT]

yes you can

Mr. Maths · 12-06-2011 02:51PM

rosskind wrote: »

If you are asked to prove something like tan(A+B), you can assume cos(A+B)=cosAcosB-sinAsinB, yeh?[FONT=Times New Roman,Times New Roman]
[/FONT]

If they did ask that, you probably could, coz it'd take too long otherwise. The one they usually ask is cos(A-B) though, coz all of the other ones are just derived from that proof.

biggaman · 12-06-2011 04:45PM

perhaps we should include some important general terms? They haven't been asked in a while and some of them are absolute ***** to figure out..

Daniel S · 12-06-2011 04:48PM

Like what?

biggaman · 12-06-2011 04:56PM

(-1)^n+1 x^2n-1 sinx
(2n-1)!

(-1)^n x^2n cos x
(2n)!

couldnt see myself getting that out without knowing it in the exam anyways

sorry not sure how well they turn out, not great at the aul formatting :P

PoisonIvy · 12-06-2011 05:15PM

Ahh the joy of leaving out Trig

Theorems/proofs for Monday

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