Question

How do you use the double angle or half angle formulas to simplify cos22.5?

Answer 1

`cos(22.5^@) =0.92388`

Explanation:

Since `22.5^@ = 1/2xx45^@` we will want to use the half angle formula for `cos`

`cos(theta/2) = +-sqrt((1+cos(theta))/2)`
`color(white)("XXX")`(we will ignore the negative version since for this example our angle falls in Quadrant I where `cos` is positive)

`cos(22.5^@) = sqrt((1+cos(45^@))/2)`

`color(white)("XXXXXX")=sqrt((1+1/sqrt(2))/2)`

`color(white)("XXXXXX")=sqrt(2+sqrt(2))/2`

(using a calculator)
`color(white)("XXXXXX")~=0.92388`