1. Find the exact value by using a half-angle identity.
sin 22.5°
Using the half angle formula you get:
sin2(θ)=12[1−cos(2θ)]
if θ=22.5° then 2θ=45°
so you get:
sin2(22.5°)=12[1−cos(45°)]
sin2(22.5°)=12[1−√2/2]=2−√24
and square root both sides:
sin(22.5°)=±√2−√24=±0.382
so
sin(22.5°)=0.382the answer is the letter D) one half times the square root of quantity two minus square root of two
2. Verify the identity.
cot x minus pi divided by two. =
-tan x
Cot(x-pi/2)=-tan(x)
sin(A − B) = sin A cos B −
cos A sin B
sin(x – pi/2) = sin x cos (pi/2)
− cos x sin (pi/2)=-cosx
cos(A − B) = cos A cos B − sin
A sin B
cos(x− pi/2) = cos x cos pi/2
− sin x sin pi/2=-sinx
Cot(x-pi/2)=cos(x-pi/2)/sin(x-pi/2)
=
(-sinx)/(-cosx)=-tanx--------------ok
