pi's Specialist Maths Questions

[ulbasour]

I need an easy way for this:


I used and then finding and expanding by making two equations:

and then expanding using Newton's generalised binomial theorem (its big)

Then collecting like terms (ie. those with and those without) and getting the answer.


Any easier ways?


EDIT:
Saw this on Wikipedia:
(Image removed from quote.)

Can someone also explain this? (I am not familiar with this notation...)


Thanks in advance!

double angle/compound angle formula.

Sin(10x)= 2sin(5x)Cos(5x) = 2Sin(3x +2x)Cos(3x +2x)= 2(Sin(3x)Cos2x +Sin(2x)Cos(3x))(Cos(3x)Cos(2x)-Sin3xSin2x) = Etc.. untill u break it down till cosx and sin x

[golden]

Instead of making a new thread for everything, I'll just make one thread. Any help with any of the question I post throughout the year will be much appreciated.

Question:


The problem I have is getting rid of .

Thanks.

May I ask where this question is from?

[pi]

Instead of making a new thread for everything, I'll just make one thread. Any help with any of the question I post throughout the year will be much appreciated.

Question:


The problem I have is getting rid of .

Thanks.

May I ask where this question is from?

School spesh teacher, he's a crazy awesome Russian who likes going (and I quote) "beyond these stupid Australian books"

[golden]

Instead of making a new thread for everything, I'll just make one thread. Any help with any of the question I post throughout the year will be much appreciated.

Question:


Haha so lots of what you do is beyond Year 12?
The problem I have is getting rid of .

Thanks.

May I ask where this question is from?

School spesh teacher, he's a crazy awesome Russian who likes going (and I quote) "beyond these stupid Australian books"

[pi]

Haha so lots of what you do is beyond Year 12?

Not much, just in his notes, he adds all of these shortcuts and stuff that aren't in books. Although for the calculus section, he's going all hardcore Russian on us

[pi]

Complex question, I just can't do it purely by hand (tech-free)




What I did:



How can I simplify this by hand (simplify the in particular)? Or is my method completely wrong?


Thanks


EDIT: I looked it up on wolframalpha, is there any way except for a continued fraction? I'm not at that level of maths yet...

[brightsky]

Let z = x + yi.
z^2 = (x+yi)^2 = x^2 + 2xy i - y^2 = (x^2 - y^2) + 2xy i.
This means (x^2 - y^2) = 8 and 2xy = -6.
etc. etc.
Hope this method is right..

[luffy]

Let z = x + yi.
z^2 = (x+yi)^2 = x^2 + 2xy i - y^2 = (x^2 - y^2) + 2xy i.
This means (x^2 - y^2) = 8 and 2xy = -6.
etc. etc.
Hope this method is right..

Your method is correct.... However, based on his "worked solution" at the question, it would seem that he wants it in polar form.

[luken93]

(1)...

(2)...







Let







 or  

Hence,

and OR

and

Apologies if I've made an error :S

[moekamo]

or you could use the formula here: http://en.wikipedia.org/wiki/Complex_number#Square_root



sgn(x) just takes the sign of x, here we take sgn(-6)=-1 since -6 is negative.

oh yea, there will be another solution 180 degrees away, i.e.

very much doubt you'd need to remember that formula in an exam though :S

[/0]

Yeah in fact for those types of questions you will probably be required to show all working, not just formulas... unless you wanna derive the formula in the exam, in which case have at it.

[pi]

...

I ended up doing it that way 

Thanks anyway guys/girls!

Just wondering if there was anyway to get rid of the ... Also, is there any quick method to realise when to switch to rcis() or when to let z=x+yi before starting the question?



New question:
Is there a simple (ie no large ugly fractions please) to:
(tech free)

When I do it (quotient rule first, and then I'm like: 'WHOA thats gonna be big!') it is pretty ugly unfortunately


Thanks!


EDIT: grammar fails

[GuessWho]

Oh god that double derivative question... Spent a few minutes doing that by hand
I saw that in the Math Quest specialist book. I don't really think there is an easy way to do it since most of the time the book would give you the simplest possible answer.

[ariawuu]

...

you mean how to work out what tan-1 (3/4) is? use the triangle.. not sure if it answers the question

[luken93]

...

you mean how to work out what tan-1 (3/4) is? use the triangle.. not sure if it answers the question

Which triangle?