Question

Use the half angle formulas to evaluate each of the following: cos(22.5 degrees) and sin(112 degrees 30')

Answer 1

\(\bf {\color{blue}{ 22.5\cdot 2\implies 45\qquad 112^o30'\implies 112.5^o\qquad 112.5\cdot 2\implies 225}}\\
cos(22.5 degrees) and sin(112 degrees 30')\\
cos(22.5^o)\implies cos\left(\frac{45^o}{2}\right)\qquad \qquad sin(112.5^o)\implies sin\left(\frac{225^o}{2}\right)\)

use the half-angle identities for those angle

Answer 2

\[\cos 2x=\cos ^{2}x-\sin ^{2}x=\cos ^{2}x-\left( 1-\cos ^{2}x \right)=2 \cos ^{2}x-1\]
\[2 \cos ^{2}x=1+\cos 2x,put x=22.5,2x=45,\cos 45=\frac{ 1 }{ \sqrt{2} }\]

Answer 3

hmmm I added som.. extra... anyhow \(\bf {\color{blue}{ 22.5\cdot 2\implies 45\qquad 112^o30'\implies 112.5^o\qquad 112.5\cdot 2\implies 225}}\\

cos(22.5^o)\implies cos\left(\frac{45^o}{2}\right)\qquad \qquad sin(112.5^o)\implies sin\left(\frac{225^o}{2}\right)\)