## anonymous 5 years ago Find the exact value of cos(22.500000000 degrees), without a calculator, using double-angle and/or half-angle identities.

1. anonymous

45/ 2

2. anonymous

cos(x/2) = cos^2(x/4) - sin^2(x/2)

3. anonymous

= 2cos^2(x/4) -1

4. anonymous

so cos^2 (x/4) = (1/2) ( 1+ cos(x/2) )

5. anonymous

sub x=90

6. anonymous

and take sqrt of both sides , note that you do get +- when you take a sqrt , but we do know that the angle is in the first quadrant and therefore the cos of the angle must be positive

7. anonymous

so cos22.5 = $\frac{1}{\sqrt{2}} ( 1+ \frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} +\frac{1}{2} = \frac{\sqrt{2} + 1 }{ 2}$

8. anonymous

wait thats wrong , should be

9. anonymous

$\frac{1}{\sqrt{2}} \sqrt{ 1 + \frac{1}{\sqrt{2}} }$