Theorems/proofs for Monday

Daniel S

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

Difference equation proof

Daniel S

polka dot wrote: »

Difference equation proof

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

polka dot

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

polka dot wrote: »

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

polka dot

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

Is that them all, then?

Daniel S

biggaman wrote: »

Is that them all, then?

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

Daniel S

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

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

cocopopsxx

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

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

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

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

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

Daniel S

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

mtb_kng wrote: »

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

They are all listed on the syllabus.

Shanee.

Difference equations are on the course?

snicket

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

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

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

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

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

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

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]
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hunii07

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

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

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

Like what?

biggaman

(-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

Ahh the joy of leaving out Trig