Trigonometric identity

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December 14th 2009, 02:09 AM
ldx2

Trigonometric identity

Can somebody solve this and give detailed description ?
I get stuck on this part 1/2 sin18..
http://img207.imageshack.us/img207/1634/firstz.jpg
December 14th 2009, 07:22 AM
Grandad

sin 18 degrees

Hello ldx2

Welcome to Math Help Forum!

Quote:

Originally Posted by ldx2

Can somebody solve this and give detailed description ?
I get stuck on this part 1/2 sin18..
http://img207.imageshack.us/img207/1634/firstz.jpg

I agree: $\sin^224^o-\sin^26^o = \tfrac12\sin 18^o$ .

There are various ways of deriving $\sin18^o$ . Here's one. And here's my version:
$\sin 18^o = \cos 72^o = 2\cos^236^o-1=2(1-2\sin^218)^2-1$

So if we put $x = \sin18^o, x$ is the root of the equation:
$x = 2(1-2x^2)^2-1$

for which $0<x<1$ .
$\Rightarrow x=2(1-4x^2+4x^4)-1$

$\Rightarrow 8x^4-8x^2-x+1=0$
Noting that $x = 1$ satisfies this equation, we can factorise:
$(x-1)(8x^3+8x^2-1)=0$
and then discard the factor $(x-1)$ , since $0<x<1$ .

We then note that $x = -\tfrac12$ is also a root. Hence $(2x+1)$ is a factor, which again we discard for the same reason; leaving us with the remaining factor:
$4x^2+2x-1=0$
the positive root of which is:
$x = \frac{\sqrt5-1}{4}=\sin18^o$
Hence $\sin^224^o-\sin^26^o = \frac{\sqrt5-1}{8}$

Grandad
December 14th 2009, 07:49 AM
ldx2

Thanks for the quick reply and solving it. EDIT: I understood until the part where you say x=-1/2 is also a root and how you found out (2x+1) is also a factor. Could you explain that part in more detail ?
December 15th 2009, 12:50 PM
Grandad

Hello ldx2

Quote:

Originally Posted by ldx2

Thanks for the quick reply and solving it. EDIT: I understood until the part where you say x=-1/2 is also a root and how you found out (2x+1) is also a factor. Could you explain that part in more detail ?

By inspection (which is a posh word for 'trial and error'), when $x = -\tfrac12, 8x^3+8x^2-1= -1+2-1=0$ . So, using the remainder theorem, $(2x+1)$ is a factor.

Grandad
December 15th 2009, 01:04 PM
ldx2

LOL 'trial and error' ! Thank you for clearing it up. Could you please take a look at my other thread where i posted 2 trig. problems !