Chapter 7: Small Angles & Approximation

Example 6

The diagram shows a rectangle ABCD where AB = 3 cm and AD = 4 cm and angle BDX = q radian.  By considering the DBDA and DXDA,
show that AX = 
3 - 4 tan q
¾¾¾¾¾
1 + ¾ tan q
If q is sufficiently small for q3 and higher powers of q to be neglected,
show that AX »  3 - (25/4)q + (75/16)q2.



Solution


tan (a - q) =  AX/4
 
\ AX = 
 4 tan (a - q)
 
= 
4 (tan a - tan q)
¾¾¾¾¾¾¾
1 + tan a tan q
 
= 
   4 (¾ - tan q)
¾¾¾¾¾¾
   1 + ¾ tan q
 
= 
  3 - 4 tan q
¾¾¾¾¾
 1 + ¾ tan q

 
 
If q is small,
AX » 
    3 - 4q
¾¾¾¾
  1 + ¾ q
 
= 
 (3 - 4q)(1 + ¾ q)-1
 
» 
 (3 - 4q)[1 - ¾ q + (¾ q)2 + ¼]
 
= 
 3 - (9/4)q + (27/16)q2 - 4q + 3q2 + ¼
 
= 
 3 - (25/4)q + (75/16)q2 + ¼

End of Chapter




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