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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: darkphoenix on April 02, 2010, 11:47:15 pm

Title: Compound and Double angle help
Post by: darkphoenix on April 02, 2010, 11:47:15 pm: Simplify each of the following expressions:

1a) $cos(3x)cos(x) + sin(3x)sin(x)$

b) $sin(A+B)cos(A-B) - cos(A+B)sin(A-B)$

Use double-angle formulas to find the exact value of:

a) sin $\frac{\pi}8$
Title: Re: Compound and Double angle help
Post by: kyzoo on April 02, 2010, 11:55:22 pm: 1.a)

cos(3x)cos(x) + sin(3x)sin(x)
= cos (3x - x) = cos 2x

b.)

sin(A+B)cos(A-B) - cos(A+B)sin(A-B)
= sin ((A+B) - (A-B))
= sin (A + B - A + B)
= sin (2B)

And argh I need latex to do the last one =/
Title: Re: Compound and Double angle help
Post by: darkphoenix on April 03, 2010, 12:12:44 am: Wait is that your final answer? If so can you show the rest of your working please?
Title: Re: Compound and Double angle help
Post by: mandy on April 03, 2010, 12:30:39 am: Quote from: darkphoenix on April 02, 2010, 11:47:15 pm
Use double-angle formulas to find the exact value of:

a) sin $\frac{\pi}8$

I'm not sure if this is what you're looking for, but:
Use $cos2x = 1 - sin^2x$

$let x = \frac{\pi}{8}$

$cos2(\frac{\pi}{8}) = 1 - sin^2(\frac{\pi}{8})$

$cos(\frac{\pi}{4}) = 1 - 2sin^2(\frac{\pi}{8})$

$\frac{\sqrt2}{2} - 1 = - 2sin^2(\frac{\pi}{8})$

$(\frac{-\sqrt2}{2} -1) / 2 = sin^2(\frac{\pi}{8})$

$\frac{-(\sqrt2 -2)}{4} = sin^2(\frac{\pi}{8})$

$\sqrt\frac{-(\sqrt2 - 2)}{4} = sin(\frac{\pi}{8})$

$sin(\frac{\pi}{8}) = \frac{\sqrt(2-\sqrt2)}{2}$

Is that right?
Title: Re: Compound and Double angle help
Post by: kyzoo on April 03, 2010, 12:43:58 am: Quote from: darkphoenix on April 03, 2010, 12:12:44 am
Wait is that your final answer? If so can you show the rest of your working please?

...that is all my working 0.o

You just use the formulas
cos (A - B) = cos A cos B + sin A sin B
sin (A - B) = sin A cos B - sin B cos A
Title: Re: Compound and Double angle help
Post by: darkphoenix on April 03, 2010, 12:49:06 am: Hm how did you get rid of the sin in the first one?

And cos in the second?
Title: Re: Compound and Double angle help
Post by: kyzoo on April 03, 2010, 12:52:37 am: Quote from: darkphoenix on April 03, 2010, 12:49:06 am
Hm how did you get rid of the sin in the first one?

And cos in the second?

cos (u - v) = cos u cos v + sin u sin v

Substitute u for 3x, and v for x

Hence you get cos (u - v) = cos (3x - x) = cos 2x

...

sin (u - v) = sin u cos v - sin u cos v

Substitute A + B for for u, and A - B for v

sin (u - v) = sin (A + B - (A - B)) = sin 2B

...

It's just a matter of learning the compound angle formulas. Once you've done that, they become 5 second questions.
Title: Re: Compound and Double angle help
Post by: darkphoenix on April 03, 2010, 03:32:56 pm: Ohh thanks alot man, wasnt thinking properly. Got it now.
Title: Re: Compound and Double angle help
Post by: coletrain on April 03, 2010, 06:12:05 pm: does compound angle and double angle formula come up in methods or just spesh?
Title: Re: Compound and Double angle help
Post by: qshyrn on April 03, 2010, 06:14:24 pm: Quote from: coletrain on April 03, 2010, 06:12:05 pm
does compound angle and double angle formula come up in methods or just spesh?
both. but more in spesh
Title: Re: Compound and Double angle help
Post by: AzureBlue on April 03, 2010, 06:54:51 pm: Quote from: coletrain on April 03, 2010, 06:12:05 pm
does compound angle and double angle formula come up in methods or just spesh?
Comes in General Maths (S) and probably MM 3/4. These formulae are awesome :)
Title: Re: Compound and Double angle help
Post by: the.watchman on April 03, 2010, 08:52:56 pm: Yeah, these formulas have slipped into the MM study design :)
I agree, they are awesome!
Title: Re: Compound and Double angle help
Post by: stonecold on April 03, 2010, 08:53:55 pm: ^they are rather awesome.

i just need to learn the tan ones now :)
Title: Re: Compound and Double angle help
Post by: AzureBlue on April 03, 2010, 09:38:58 pm: Quote from: the.watchman on April 03, 2010, 08:52:56 pm
Yeah, these formulas have slipped into the MM study design :)
I agree, they are awesome!
I don't think they're in 1/2... 1/2 sucks. Just sin, cos, tan graphs I think and the circle.
Title: Re: Compound and Double angle help
Post by: the.watchman on April 03, 2010, 09:49:28 pm: Quote from: AzureBlue on April 03, 2010, 09:38:58 pm
Quote from: the.watchman on April 03, 2010, 08:52:56 pm
Yeah, these formulas have slipped into the MM study design :)
I agree, they are awesome!
I don't think they're in 1/2... 1/2 sucks. Just sin, cos, tan graphs I think and the circle.

That's right, 1/2 is rather boring
3/4 is better, but still boring :P
Title: Re: Compound and Double angle help
Post by: qshyrn on April 03, 2010, 09:49:59 pm: Quote from: AzureBlue on April 03, 2010, 09:38:58 pm
Quote from: the.watchman on April 03, 2010, 08:52:56 pm
Yeah, these formulas have slipped into the MM study design :)
I agree, they are awesome!
I don't think they're in 1/2... 1/2 sucks. Just sin, cos, tan graphs I think and the circle.
dont think theyre in the study design for 1/2, but i had to learn them in yr 11
Title: Re: Compound and Double angle help
Post by: TrueTears on April 03, 2010, 09:51:17 pm: expressions like these are cool
Title: Re: Compound and Double angle help
Post by: AzureBlue on April 04, 2010, 08:32:52 am: Quote from: TrueTears on April 03, 2010, 09:51:17 pm
expressions like these are cool
+1
The best thing I've found in VCE Maths so far :)