De Moivre's Theorem (1 Viewer)

Pedro123

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Not sure how to solve the following:

I can get it down to proving:

But don't know where to go from there. I'm almost certain you need to use:

But I'm not sure how I'm supposed to use that. Doing it with induction also is a bit of a pain in the inductive step then.
For reference, this is Cambridge Maths Year 12 Extension 2 3A Question 17. I found this much harder than the enrichment problems, which leads me to believe I'm fucking this up somehow, but I don't know how.

Thanks
 

B1andB2

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Have you tried getting rid of the one by using double angle?

Eg first bracket
(1 + cos^a/2 -sin^a/2 + 2i Sin a/2 cos a/2)^k
And then simplify 1- sin^ a/2 to be cos^ a/2

so you get 2cos^2 (a/2) + ...

idk just a thought
 

idkkdi

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Not sure how to solve the following:

I can get it down to proving:

But don't know where to go from there. I'm almost certain you need to use:

But I'm not sure how I'm supposed to use that. Doing it with induction also is a bit of a pain in the inductive step then.
For reference, this is Cambridge Maths Year 12 Extension 2 3A Question 17. I found this much harder than the enrichment problems, which leads me to believe I'm fucking this up somehow, but I don't know how.

Thanks
someone's learning ahead.
 

mani88

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Not sure how to solve the following:

I can get it down to proving:

But don't know where to go from there. I'm almost certain you need to use:

But I'm not sure how I'm supposed to use that. Doing it with induction also is a bit of a pain in the inductive step then.
For reference, this is Cambridge Maths Year 12 Extension 2 3A Question 17. I found this much harder than the enrichment problems, which leads me to believe I'm fucking this up somehow, but I don't know how.

Thanks
Use Double angle formulas to get rid of the 1 in the LHS and then start from there
 

B1andB2

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This is what I’ve been doing

I don’t know how to finish off the last line, but most of it is there
Sorry it’s so messy

also just realised I forgot to write brackets to show it’s all to the power of k whoops just imagine I did
 

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Trebla

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Not sure how to solve the following:

I can get it down to proving:

But don't know where to go from there. I'm almost certain you need to use:

But I'm not sure how I'm supposed to use that. Doing it with induction also is a bit of a pain in the inductive step then.
For reference, this is Cambridge Maths Year 12 Extension 2 3A Question 17. I found this much harder than the enrichment problems, which leads me to believe I'm fucking this up somehow, but I don't know how.

Thanks
As a hint



 

Pedro123

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Thanks guys - turns out I just pulled a bit of a dumb :)

Out of curiosity, does anyone have any ideas to prove the sum of cosine expression I thought I needed?
 

mani88

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Yep I see it now
sorry that was a bit incorrect, i just fixed it up
 

Trebla

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Out of curiosity, does anyone have any ideas to prove the sum of cosine expression I thought I needed?
Just expand the RHS using compound angles and it works out. You actually don't even need to memorise this because it's in the reference sheet.
 

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