Answer
$F\left( \alpha \right) = \frac{2}{\pi }\sin \frac{{\pi \alpha }}{2} + \frac{2}{\pi }$
Work Step by Step
$$\eqalign{
& F\left( \alpha \right) = \int_{ - 1}^\alpha {\cos \frac{{\pi \theta }}{2}} d\theta \cr
& {\text{Integrate}} \cr
& F\left( \alpha \right) = \frac{2}{\pi }\left[ {\sin \frac{{\pi \theta }}{2}} \right]_{ - 1}^\alpha \cr
& F\left( \alpha \right) = \frac{2}{\pi }\left[ {\sin \frac{{\pi \alpha }}{2} - \sin \frac{{\pi \left( { - 1} \right)}}{2}} \right] \cr
& F\left( \alpha \right) = \frac{2}{\pi }\left[ {\sin \frac{{\pi \alpha }}{2} + 1} \right] \cr
& F\left( \alpha \right) = \frac{2}{\pi }\sin \frac{{\pi \alpha }}{2} + \frac{2}{\pi }{\text{ }}\left( {{\text{accumulation function}}} \right) \cr
& \cr
& \left( {\text{a}} \right)F\left( { - 1} \right) \cr
& F\left( { - 1} \right) = \frac{2}{\pi }\sin \frac{{\pi \left( { - 1} \right)}}{2} + \frac{2}{\pi } \cr
& F\left( { - 1} \right) = - \frac{2}{\pi } + \frac{2}{\pi }{\text{ }} \cr
& F\left( { - 1} \right) = 0,{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& \left( {\text{b}} \right)F\left( 0 \right) \cr
& F\left( 0 \right) = \frac{2}{\pi }\sin \frac{{\pi \left( 0 \right)}}{2} + \frac{2}{\pi } \cr
& F\left( 0 \right) = \frac{2}{\pi },{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& \left( {\text{c}} \right)F\left( {\frac{1}{2}} \right) \cr
& F\left( {\frac{1}{2}} \right) = \frac{2}{\pi }\sin \frac{{\pi \left( {1/2} \right)}}{2} + \frac{2}{\pi } \cr
& F\left( {\frac{1}{2}} \right) = \frac{2}{\pi }\left( {\frac{{\sqrt 2 }}{2}} \right) + \frac{2}{\pi } \cr
& F\left( {\frac{1}{2}} \right) = \frac{{\sqrt 2 }}{\pi } + \frac{2}{\pi } \cr
& F\left( {\frac{1}{2}} \right) = \frac{{2 + \sqrt 2 }}{\pi },{\text{ }}\left( {{\text{Graph shown below}}} \right) \cr
& \cr
& {\text{Graphs}} \cr} $$
