Chapter 7: Small Angles & Approximation

In a more formal approach to small values of q, areas are considered:
 
Area of triangle AOB
  = ½ r2 sin q
Area of sector AOB
  = ½ r2q
Area of triangle AOD
  = ½(OA)(AD)
 = ½(r)(r tan q)
 = ½ r2 tan q


Area of triangle AOB < 
Area of sector AOB
  < Area of triangle AOD
  
½ r2 sin q < 
½ r2q
  < ½ r2 tan q
  
sin q < 
q
  < tan q
  
sin q
¾¾
  < 
sin q
q
¾¾
sin q
tan q
 < 
¾¾
sin q
  
1 < 
q
¾¾
sin q
1
 <  ¾¾
cos q

 
 
As q ® 0, cos q ® 1 Þ 
1
¾¾
cos q
 ® 1.
\  
q
¾¾
sin q
 ® 1
i.e. 
lim
q ® 0
q
¾¾
sin q
 = 1, which implies 
q
¾¾
sin q
 » 1.

 
Þ 
For small Ð q , sin q » q.




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