Question

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

Answer 1

45/ 2

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

sub x=90

Answer 6

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

Answer 7

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

Answer 8

wait thats wrong , should be

Answer 9

\[\frac{1}{\sqrt{2}} \sqrt{ 1 + \frac{1}{\sqrt{2}} }\]