Use the half formula to find the exact value of the expression sin 22.5
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.
My options are
-1/2 sqrt2 + sqrt2
1/2 sqrt2 + sqrt-2
-1/2 sqrt2 - sqrt2
1/2 sqrt2 + sqrt2
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.
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
To find the exact value of the expression sin 22.5 using the half-angle formula, we need to use the formula:
sin (θ/2) = √[(1 - cos θ) / 2]
In this case, θ = 45 degrees. So we can rewrite the formula as:
sin (45/2) = √[(1 - cos 45) / 2]
Now, let's find the values of cos 45 and plug them into the formula:
cos 45 = 1 / √2 = √2 / 2
Substituting the value of cos 45 back into the formula:
sin (45/2) = √[(1 - √2 / 2) / 2]
To simplify the expression further, we can rationalize the denominator:
sin (45/2) = √[(2 - √2) / 4]
Now, we can simplify the expression by multiplying the numerator and denominator by √2:
sin (45/2) = (√2 * √(2 - √2)) / (2 * √2)
After simplifying:
sin (45/2) = (√(4 - 2√2)) / 2√2
Therefore, the exact value of sin 22.5 using the half-angle formula is (√(4 - 2√2)) / 2√2.