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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: jasopan on April 17, 2010, 05:56:26 pm

Title: Evaluate cos(pi/8)
Post by: jasopan on April 17, 2010, 05:56:26 pm
Evalaluate cos(pi/8) using half angle Formulae leaving answer in exact values


Thanks!
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 17, 2010, 06:01:31 pm


Use half-angle formula:

In this case:

So



So:

Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 17, 2010, 06:05:00 pm
^^ Yup, got the sin cos ones, but now I'm stuck on the tan
I can't seem to use the formula, or should i do
tanx=sinx/cosx and then simplify?

And also is there a way to solve the sin(pi/8) using sinX=2sin(x/2)cos(X/2)
Thanks again
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 09:59:00 pm
there is another way to evaluate sin(pi/8) and cos(pi/8) using complex numbers if anyone is interested. In fact, it make a a rather good question. It's in the essentials book, pg 168, Q8
Title: Re: Evaluate cos(pi/8)
Post by: physics on April 17, 2010, 10:35:19 pm


Use half-angle formula:

In this case:

So



So:


sucky uni maths stuff :( its so hard D:
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 10:46:35 pm
lol, uni maths stull is AWESOME, and that not uni maths, it's spech
Title: Re: Evaluate cos(pi/8)
Post by: physics on April 17, 2010, 10:48:30 pm
lol, uni maths stull is AWESOME, and that not uni maths, it's spech
we're doing this at uni maths D: i dont do spesh so i not know haha
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 10:51:32 pm
wtf, i just realised that. How are you doing uni maths without spech? not only how but why???
Title: Re: Evaluate cos(pi/8)
Post by: physics on April 17, 2010, 10:53:12 pm
wtf, i just realised that. How are you doing uni maths without spech? not only how but why???
how because they let me..i dont know y every1 says u have to do spesh b4 uni maths :S but they let me
WHY? so i can drop a subject and profit in free hours haha
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 17, 2010, 11:01:44 pm
there is another way to evaluate sin(pi/8) and cos(pi/8) using complex numbers if anyone is interested. In fact, it make a a rather good question. It's in the essentials book, pg 168, Q8

Can you post up the proof? :)
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 11:05:54 pm
reli? dammit i hate latex have never used it cos i ceebes
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 17, 2010, 11:07:20 pm
Quick outline? Just to quench my curiosity?  :P
Title: Re: Evaluate cos(pi/8)
Post by: TrueTears on April 17, 2010, 11:08:14 pm
latex is the bomb, once u get into uni latex is ur best friend for maths xD might as well learn the skill now
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 11:11:36 pm
i'll take you thu the logic you'll learn more by doing the numbers youself(and it'll save me time lol)

ok

1)It's the same logic as questions 3 and 4. Let z^2 be 1+i and let z be a+bi
2) square (a+bi) and let it equal 1+i, solve for a and b

part ii) you should be able to do, it's pretty standard, just basic using de moirves

part b)

1) expand the and from part aii into carterian form, not symplyfing cos(pi/8) and sin(pi/8)
2) equate coefficents

Hope that helps:) Ask if you need more help
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 17, 2010, 11:14:09 pm
Lol I don't have the book.  :(
Title: Re: Evaluate cos(pi/8)
Post by: tram on April 17, 2010, 11:16:44 pm
dammit, TT you have e-books of everything, do you have essentials???
Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 05:28:54 pm
Hey can someone re-explain the


And one more;
Show that:

tA  :)
Title: Re: Evaluate cos(pi/8)
Post by: Yitzi_K on April 18, 2010, 06:01:02 pm
And one more;
Show that:

tA  :)















as required.
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 18, 2010, 06:12:23 pm


Use the identity .

The rest is trivial.
Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 06:33:11 pm
^ Whoa, my book doesn't even have that identity ...  >:(
Gotta google some now  :-\

@Yitzi_K yes! got it! thanks alotalotalotalot

Another one since everyone's so helpful :D

Simplify    divide   
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 18, 2010, 06:41:29 pm
You can derive the formula from this image:

http://upload.wikimedia.org/wikipedia/commons/2/21/Weierstrass_substitution.png

:D



Look at , then use as normal.
Title: Re: Evaluate cos(pi/8)
Post by: TrueTears on April 18, 2010, 06:45:35 pm
dammit, TT you have e-books of everything, do you have essentials???
yeah i do...
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 18, 2010, 07:02:57 pm






Let .







So:













:)
Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 07:06:14 pm
^ Holy crap. Is it that hard?
I tried to use tanx=sinx/cosx to get tanx as the book didn't have the right formula and thats what i got stuck on :S

Thanks again brightsky, this is gold
Title: Re: Evaluate cos(pi/8)
Post by: superflya on April 18, 2010, 07:17:54 pm
or u could use the double angle formula



let

  and note

rearrange to get a quadratic    

solve and u get

as

Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 09:01:45 pm
What does it mean if they go 'if x is an acute angle' ? does that mean its only in the first quadrant ? so all sin cos tan are positive?

And just to confirm if domain is pi > x > pi/2, then domain of x/2 is (pi/4, pi/2)
Title: Re: Evaluate cos(pi/8)
Post by: brightsky on April 18, 2010, 09:03:42 pm
Yep. Acute means for angle , hence in the first quadrant.

Yes for your second question as well (I'm pretty sure).
Title: Re: Evaluate cos(pi/8)
Post by: superflya on April 18, 2010, 09:04:49 pm




Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 09:16:08 pm

for     

Is the domain written;

      or     
Title: Re: Evaluate cos(pi/8)
Post by: m@tty on April 18, 2010, 09:22:59 pm
The second one.

is two statements combined.

and

So and .

Or .
Title: Re: Evaluate cos(pi/8)
Post by: jasopan on April 18, 2010, 09:27:43 pm
Yup, all clear now, thanks for clearing that up
Edit sorry another one, if x is an acute angle, and then we are asked to find sin(x/2) are all tanx/2 sinx/2 cosx/2 still positive? do we have to state the new domain is pi/4>x>0?
EDIT2 nvm, i understand it now ;D