I had this on a test:
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
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?)
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) )
I was thinking clearly, silly mistake that's for sure , thanks
To find the exact value of sin 22.5 degrees using the half-angle formula, we can start by applying the half-angle formula for sine:
sin(θ/2) = √((1 - cos θ)/2)
Now, let's substitute θ = 45 degrees into this formula since 45 degrees is twice the angle we need, which is 22.5 degrees:
sin(45/2) = √((1 - cos 45)/2)
To find the value of cos 45, we can use the fact that cos(45 degrees) = √2 / 2:
sin(45/2) = √((1 - √2/2)/2)
Simplifying further, we get:
sin(45/2) = √((2 - √2) / 4)
Now, let's rationalize the denominator by multiplying both the numerator and the denominator by √2 to eliminate the square root from the denominator:
sin(45/2) = √((2 - √2) / 4) * (√2/√2)
sin(45/2) = √((2 - √2) * √2 / (4 * √2))
sin(45/2) = √((2√2 - 2) / 4√2)
Next, we can simplify the numerator:
sin(45/2) = √((2√2 - 2) / 4√2) = √((√2(2 - √2)) / (4√2))
Since there is a common factor of √2 in both the numerator and the denominator, they cancel out:
sin(45/2) = √((2 - √2)/4)
Finally, we simplify the expression further by dividing both the numerator and the denominator by 2:
sin(45/2) = √((2 - √2)/4) = √((1 - √2/2)/2) = √((1 - cos 45)/2)
Thus, the exact value of sin 22.5 degrees using the half-angle formula is:
sin(22.5 degrees) = √((1 - cos 45)/2)