Trigonometry (Continued)


   
 
Projection Formula
Theorem :
 
In any triangle ABC, with sides a = BC, b = CA and c = BA, then prove that a = b cos C + c cos B, b = c cos A +    a cos C,  c = a cos B + b cos A
 
 
                 Fig (i)
 
 
                  Fig (ii)
 
 
                          Fig (iii)
 
Proof:
 
Let ABC be any triangle and let AD be drawn perpendicular to BC.
 
(AD = h = altitude of triangle)
 
In figure (i), we have
 
BD = x, DC = y
 
 
From right-angled triangle ADB, we have
 
 
From right-angled triangle ADC, we have
 
 
Adding (ii) and (iii), we get
 
x + y = cos B + b cox C …. (iv)
 
From (i) and (iv), we have
 
 
From figure (ii), we have
 
BD = x, DC = y
 
 
In D ADB,
 
 
In D ADC,
 
 
 
 
 
From fig (iii),
 
 
 
Hence, a = b cos C + c cos B.
 
Similarly, other three results can be proved.
 
  
     
   
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