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Evaluation of the Coefficientssinx2sin xFigure 7.5. Graphical proof of∫02π( sinmt) 2 dt=πcosx2cos xFigure 7.6. Graphical proof ofTo evaluate any coefficient in (7.10), say , we multiply both sides of (7.10) by sin2t. Then,ft () sin2t=b 2∫02π( cosmt) 2 dt=1--a2 0 sin2t+ a 1 costsin2t+ a 2 cos2tsin2t+ a 3 cos 3tsin2t+ a 4 cos4tsin2t+b 1 sintsin2t+ b 2 ( sin2t) 2 + b 3 sin 3tsin2t+ b 4 sin4tsin2t+ …π…Next, we multiply both sides of the above expression byto 2π . Then,dt, and we integrate over the period 0∫02πft () sin2tdt1= --a2 0 ∫sin2td t + a 1 ∫costsin2td t + a 2 ∫cos2tsin2tdt02π∫02π+ a 3 cos3tsin2tdt+ …02π+ b 1 ∫sintsin 2t d t + b 2 ∫( sin2t) 2 d t + b 3 ∫sin3tsin2tdt + …02π02π02π02π(7.11)Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth EditionCopyright © Orchard Publications7−5