# Find the exact value of Cos 11.25 degrees

• Dec 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?
• Dec 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 $\displaystyle \cos(4\theta)}~?$.

Because $\displaystyle 45^o=4(11.25)^o.$
• Dec 9th 2012, 10:53 AM
richard1234
Re: Find the exact value of Cos 11.25 degrees
11.25 degrees = pi/16

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

Then just apply this half-angle formula a couple times to get your answer.
• Dec 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?
• Dec 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 ...

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

now let $\displaystyle x = 45^\circ$
• Dec 9th 2012, 11:40 AM
Eraser147
Re: Find the exact value of Cos 11.25 degrees
^ Makes a lot more sense. Thanks.