Answer
(a) $\frac{1}{2}$
(b) $\frac{\sqrt 2+\sqrt 6}{4}$
(c) $\frac{\sqrt {2-\sqrt 3}}{2}$
Work Step by Step
(a) Use the Product-Sum formula, we get $sin8^\circ co22^\circ+cos8^\circ sin22^\circ=sin(8^\circ+22^\circ)=sin30^\circ=\frac{1}{2}$
(b) Use the Addition Formula, $sin75^\circ=sin(45^\circ+30^\circ)=sin45^\circ cos30^\circ+cos45^\circ sin30^\circ=\frac{\sqrt 2}{2}\times\frac{\sqrt 3}{2}+\frac{\sqrt 2}{2}\times\frac{1}{2}=\frac{\sqrt 2+\sqrt 6}{4}$
(c) Use the Half=Angle Formula, $sin\frac{\pi}{12}=\sqrt {\frac{1-cos\frac{\pi}{6}}{2}}=\sqrt {\frac{1-\frac{\sqrt 3}{2}}{2}}=\sqrt {\frac{2-\sqrt 3}{4}}=\frac{\sqrt {2-\sqrt 3}}{2}$
