Find the exact value of Cos 11.25 degrees

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December 9th 2012, 10:42 AM
Eraser147

Find the exact value of Cos 11.25 degrees

How do I find the exact value without any decimals?
December 9th 2012, 10:50 AM
Plato

Re: Find the exact value of Cos 11.25 degrees

Quote:

Originally Posted by Eraser147

How do I find the exact value without any decimals?

Can you find an easy expression for $\cos(4\theta)}~?$ .

Because $45^o=4(11.25)^o.$
December 9th 2012, 10:53 AM
richard1234

Re: Find the exact value of Cos 11.25 degrees

11.25 degrees = pi/16

$\cos 2\theta = 2 \cos^2 \theta - 1 \Rightarrow \cos \theta = \sqrt{\frac{1 + \cos 2\theta}{2}}$ (in the domain that we're interested in)

Then just apply this half-angle formula a couple times to get your answer.
December 9th 2012, 11:18 AM
Eraser147

Re: Find the exact value of Cos 11.25 degrees

So it would be Sqrt 1 + (2)11.25 divided by 2?
December 9th 2012, 11:27 AM
skeeter

Re: Find the exact value of Cos 11.25 degrees

Quote:

Originally Posted by Eraser147

So it would be Sqrt 1 + (2)11.25 divided by 2?

no ...

$\cos\left(\frac{x}{4}\right) = \sqrt{\frac{1 + \cos\left(\frac{x}{2}\right)}{2}} = \sqrt{\frac{1 + \sqrt{\frac{1+\cos{x}}{2}}}{2}}$

now let $x = 45^\circ$
December 9th 2012, 11:40 AM
Eraser147

Re: Find the exact value of Cos 11.25 degrees

^ Makes a lot more sense. Thanks.

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