Use the half formula to find the exact value of the expression sin 22.5

Answer 1

one of the formulas for cos 2A is

cos 2A = 1 - 2sin^2 A

so if you let A = 22.5, then 2A = 45 and you should know the basic trig ratios for the 45º angle

cos 45 = 1 - 2sin^2 22.5

sin^2 22.5 = (1 - (√2)/2)/2

take the square root again

sin 22.5 = √[(1 - (√2)/2)/2]

and there is your "exact" value

check it with a calculator.

Answer 2

My options are

-1/2 sqrt2 + sqrt2
1/2 sqrt2 + sqrt-2
-1/2 sqrt2 - sqrt2
1/2 sqrt2 + sqrt2

Answer 3

none of those answers make any sense

"-1/2 sqrt2 + sqrt2 " ----> that is sin 45º
"1/2 sqrt2 + sqrt-2 " ----> sqrt-2 is undefined
"-1/2 sqrt2 - sqrt2 " ----> that would be a negative result, the sin 22.5 is definitely positive
"1/2 sqrt2 + sqrt2" ----> sine of any angle cannot be greater than 1, sqrt2 is already over 1

by calculator
sin 22.5 = .3826834
evaluating my answer gives .3826834, I don't know why you doubt me.

Answer 4

I don't doubt you it doesn't make sense to me either but those are the choices I'm given let me look into this a little more and I'll re-post if need be