Chapter 7: Small Angles & Approximation

7.3  Worked Examples

Example 1

Given that q is small, find an approximation for the following expressions:
(a) sin 2q tan 2q
¾¾¾¾¾¾
1 - cos q		         (b) sin 3q
¾¾¾¾¾
1 + cos 2q


Solution

(a)  sin 2q tan 2q
¾¾¾¾¾¾
1 - cos q		  »  (2q)(2q)
¾¾¾¾¾¾
1 - (1 - ½ q2)
 
 = 
4q2
¾¾¾
½ q2
  
 = 8




(b) 
sin 3q
¾¾¾¾¾
1 + cos 2q
  »              3q
¾¾¾¾¾¾¾
 1 + [1 - ½(2q)2]
 
 =        3q
¾¾¾¾
 2(1 - q2)
 
 =  3
- q(1 -q2)-1
2 
 
 =  3
- q(1 + q2 + ¼)
2
 
 »  3
- q   (assuming q3 is small
2       enough to ignore)




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