# FP2: De Moivre's theorem and it's applications Watch

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So I just finished this chapter (AQA) and my brain is absolutely fried. There is so much to take in and it's so dense with information.

Does anybody have any helpful resources to help with it? I have used Examsolutions which is nice but I'm looking for more.

Does anybody have any helpful resources to help with it? I have used Examsolutions which is nice but I'm looking for more.

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#3

De Moivre's formula? I thought De Moivre's formula was a direct result of Euler's exponential representation of complex numbers.

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#4

(Original post by

my brain is absolutely fried.

**darkforest**)my brain is absolutely fried.

Anyway, building from De Moivre's Theorem.

You learnt the expansions of and in powers of and .

Let's skip ahead and assume you have proved the following,

Now with Edexcel, there's a chapter called Integration by Reduction (in FP3), this is usually integrating a function involving usually being the power and is high. (IMO Integration by Reduction is faster but it's worth mentioning.)

You can integrate the function since you've expanded it,

i.e.

Alternatively, you'd use Integration by Reduction by setting the power as n, i.e. then use integration by parts etc. (I won't get into it as this isn't related to complex numbers but it's the idea of integrating a function to a high power as shown).

Pretty much the only new thing I've learnt after A Level regarding De Moivre's Theorem. The rest is Euler's exponential representation.

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#5

(Original post by

(IMO Integration by Reduction is faster but it's worth mentioning.)

**ManLike007**)(IMO Integration by Reduction is faster but it's worth mentioning.)

**odd**, the fastest method is to rewrite as:

and then integrate by recognition.

e.g.

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#6

(Original post by

So I just finished this chapter (AQA) and my brain is absolutely fried. There is so much to take in and it's so dense with information.

Does anybody have any helpful resources to help with it? I have used Examsolutions which is nice but I'm looking for more.

**darkforest**)So I just finished this chapter (AQA) and my brain is absolutely fried. There is so much to take in and it's so dense with information.

Does anybody have any helpful resources to help with it? I have used Examsolutions which is nice but I'm looking for more.

and the two main applications are:

you can expand binomially to get expressions for cos kx and sin kx in terms of cos x and sin x.

Conversely, setting z = cos x + i \sin x, then z+(1/z) = 2 cos x and z-1/z = 2i sin x; you can use this and expand (z+1/z)^k (or (z-1/z)^k binomially to get expressions for cos^k x and sin^k x as a linear combination of basic trig expressions (e.g. cos^3 x = (cos 3x + cos x)/4), which is handy for reducing to something that's easy to integrate.

I'm not sure what else you think you need to know?

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#7

(Original post by

I know this is taking things even more off topic, but FWIW, to integrate where n is

and then integrate by recognition.

e.g.

**DFranklin**)I know this is taking things even more off topic, but FWIW, to integrate where n is

**odd**, the fastest method is to rewrite as:and then integrate by recognition.

e.g.

(Apologies to OP for swaying this off even more)

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Thanks for all of the replies!

It seems like I was really just struggling with the concept of the ‘n nth roots of complex numbers’.

I did a bit more work on it and I seem to get it now.

It seems like I was really just struggling with the concept of the ‘n nth roots of complex numbers’.

I did a bit more work on it and I seem to get it now.

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#9

(Original post by

I'm assuming this also works for for odd values of n. It's interesting but I believe there is a doable limit you'd use this 'trick' as a power of means any number, I mean if you had , you know, it's not really fun to expand now is it? (Unless you can do Binomial expansion in your head then fair play to you sir)

**ManLike007**)I'm assuming this also works for for odd values of n. It's interesting but I believe there is a doable limit you'd use this 'trick' as a power of means any number, I mean if you had , you know, it's not really fun to expand now is it? (Unless you can do Binomial expansion in your head then fair play to you sir)

So integral is: S - 4S^3/3 +6S^5/5 -4S^7/7 + S^9 / 9, where S = sin x.

Which I literally wrote down.

Note that using a reduction formula is also not really fun when n is large (except for certain special cases where the limits mean it works out unusually nicely).

There are still times where you want to use a reduction anyhow (particularly when you want to examine the behaviour as n->infinity), but for a standard question, it saves a bit of time.

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#10

**DFranklin**)

I know this is taking things even more off topic, but FWIW, to integrate where n is

**odd**, the fastest method is to rewrite as:

and then integrate by recognition.

e.g.

Attached to my post is how a similar integral would appear on an exam paper.

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#11

(Original post by

PRSOM for the nifty method, but did you really think that A Level maths exams would afford you opportunities to use whichever method you wished? If that were the case I'd probably have used modular arithmetic instead of induction for all the divisibility questions in FP1.

Attached to my post is how a similar integral would appear on an exam paper.

**I hate maths**)PRSOM for the nifty method, but did you really think that A Level maths exams would afford you opportunities to use whichever method you wished? If that were the case I'd probably have used modular arithmetic instead of induction for all the divisibility questions in FP1.

Attached to my post is how a similar integral would appear on an exam paper.

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#12

(Original post by

Well once part a) has already been done, the quickest way is to use it instead of using DFranklin's method

**Notnek**)Well once part a) has already been done, the quickest way is to use it instead of using DFranklin's method

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#13

(Original post by

Attached to my post is how a similar integral would appear on an exam paper.

**I hate maths**)Attached to my post is how a similar integral would appear on an exam paper.

But yeah, if they tell you to use a stupid method, you have to use a stupid method. (And I don't think there's much doubt that there are at least 2 methods *significantly* quicker than the one they tell you to use).

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#14

(Original post by

But yeah, if they tell you to use a stupid method, you have to use a stupid method. (And I don't think there's much doubt that there are at least 2 methods *significantly* quicker than the one they tell you to use).

**DFranklin**)But yeah, if they tell you to use a stupid method, you have to use a stupid method. (And I don't think there's much doubt that there are at least 2 methods *significantly* quicker than the one they tell you to use).

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#15

(Original post by

While this is true for a lot of A Level questions, I think for the one posted above it's really testing that a student can apply De Moivre's theorem. The integration is just a "nice" (in their opinion) follow up and isn't really testing that much integration ability.

**Notnek**)While this is true for a lot of A Level questions, I think for the one posted above it's really testing that a student can apply De Moivre's theorem. The integration is just a "nice" (in their opinion) follow up and isn't really testing that much integration ability.

[More generally, I think the (z+1/z)^k trick is used very rarely outside of this kind of A-level question, so I find it frustrating that they try to pretend it's actually "equivalently useful" to the (c+is)^k trick. I'd guess I've used (c+is)^k ten times more often that the (z+1/z) trick with real problems.]

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