I had this on a test:

Question

Use the half-angle formula to find the exact value of the expresseion:
sin 22.5 degrees

my answer - 1/2 sqrt2 - sqrt 2 was wrong

Answer 1

22.5 is half of 45º, those trig ratios are known
using cos 2x = 1 - 2sin^2 x
and knowing that cos 45º = √2/2

cos 45º = 1 - 2 sin^2 22.5º
√2/2 = 1 - 2sin^2 22.5
sin^2 22.5 = (1-√2/2)/2

sin 22.5 = √[(1-√2/2)/2]

This probably could be simplified a bit, but it is the "exact" answer

( you should have realized that sin 22.5 could not possibly be negative, like your answer is
Why didn't you use your calculator to check your answer?)

Answer 2

cos45= - sin^2 22.5 +cos^2 22.5
=1 - 2sin^2 22.5

sin 22.5= sqrt (-1/2 cos45 +1/2 )
= sqrt (1/2 - 1/(2sqrt2) )

Answer 3

I was thinking clearly, silly mistake that's for sure , thanks