Question

Find the exact value of cos 105 by using a half-angle formula.​

Answer 1

The half-angle formula for cosine is cos(x/2) = ±sqrt((1 + cos(x))/2).

Using this formula, we can find cos(105) as follows:

cos(105) = cos(210/2)
cos(105) = sqrt((1 + cos(210))/2)
cos(105) = sqrt((1 - sqrt(3)/2)/2)
cos(105) = sqrt(2 - sqrt(3))/2

Therefore, the exact value of cos(105) using the half-angle formula is sqrt(2 - sqrt(3))/2.