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Prove. Prove that

 

ole.gif  


can be written as


ole1.gif  

 

where ω0 = 2π/T and

 


             ole2.gif


or as


ole3.gif  

 

where


             ole4.gif



Proof. The expression ancos nω0t + bnsin nω0t in 1) can be written as


ole5.gif  


ole6.gif

Let us now define the angles θn and ole7.gif in accordance with the triangle shown in Fig. 1 and set


             ole8.gif  


With this notation 4) can be written as


ole9.gif


Using a formula from trigonometry that states that


            cos (A - B) = cos A cos B + sin A sin B


we can write 5) as  


ole10.gif

 

Using this result, 1) can be written as


ole11.gif

 

which is the result required for 2) above. To obtain result 3) we note that 4) can also be written as  


ole12.gif


Using a formula from trigonometry that states that


            sin (A + B) = sin A cos B + sin B cos A


we can write 8) as  


ole13.gif


Thus 1) can be written as

ole14.gif

 

which is the result required for 3) above.


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