using an appropriate half angle identity what is the exact value of cosine22.5 degrees?

Question

half angle : cosine theta divide by 2= plus or minus the square root of (1-cosine theta divided by 2)
***the answer is supposed to be the sqare root of (2+ (the sqare root of 2) divided by 2) i am lost in the process???

Answer 1

cos 22.5 = cos (45/2)
cos 45 = sqrt(2)/2 = 2 cos^2 (22.5) - 1
=> cos^2(22.5) = sqrt(2)/4 + 1/2 = (sqrt(2) + 2)/4
=> cos(22.5) = sqrt ( sqrt(2 ) + 2) /2 (22.5 < 90 thus cos 22.5 > 0)

Answer 2

Like you said, the half angle identity goes as follows:

cos^2(x) = (1 + cos2x)/2

Therefore,

cos^2(22.5) = (1 + cos[2*22.5])/2

cos^2(22.5) = (1 + cos[45])/2

BUT, we know the cosine of 45; it is equal to sqrt(2)/2, so we can calculate it.

cos^2(22.5) = (1 + sqrt(2)/2)/2

We can get rid of that complex fraction by multiplying top and bottom by 2. This will get rid of all fractions within fractions.

cos^2(22.5) = (2 + sqrt(2))/4

However, we want cos(22.5), NOT cos^2(22.5). What we have to do is take the square root of both sides.

cos(22.5) = +/- sqrt (2 + sqrt(2))/2

However, we know what quadrant 22.5 degrees lies in (the first quadrant), and in the first quadrant, cosine is positive. Therefore, we discard the negative result.

cos(22.5) = sqrt(2 + sqrt(2))/2