36 Solve trigo equation

Prévia do material em texto

Solve a sin x + b cos x = c
Yue Kwok Choy
Question
Solve : 	12 sin x + 5 cos x = 4
Method 1
Construct a triangle OPQ with 
 .
( 	( ( 67.380o .
Also,		
Now,		12 sin x + 5 cos x = 4
		(13 sin () sin x + (13 cos () cos x = 4
		13 (cos x cos ( + sin x sin () = 4
		13 cos (x – () = 4 
		
		
		
	(	
	,where n is an integer .	….	(1)
						
Method 2
Construct a triangle OPQ with 
 .
( 	( ( 22.620o .
Also,		
Now,		12 sin x + 5 cos x = 4
		(13 cos () sin x + (13 sin () cos x = 4
		13 (sin x cos ( + cos x sin () = 4
		13 sin ( x + () = 4
		
		
		
	, where n is an integer .				….	(2)
			
To show that the solution set (2) is the same as the solution set (1), we need to divide the solution into odd and even cases : 
(a) 	n = 2m , 	
(b)	n = 2m + 1, 	
, where m is an integer.
Method 3		(t – method)
Let 
 .
Now, the equation :		12 sin x + 5 cos x = 4
					
					
					
					
(	
(	
(	
 , where n is an integer .		….	(3)
Method 4		(unsatisfactory)
12 sin x + 5 cos x = 4
				….	(4)
Method 4 is unsatisfactory since solution set (4) gives a larger solution set than (1), (2) or (3).
Readers please explain. What solutions should be rejected ? How to reject ?
Explanation
Squaring equation creates roots! That is why Method 4 is not recommended.
Finding the redundant roots is tedious:
In (4),
(a)	When	
.
	(i)	n = 2m,		
	(ii)	n = 2m + 1,	
(b)	When	
(i)	n = 2m,		
(ii)	n = 2m + 1,	
Therefore (4) is equivalent to :
	
(4.2) and (4.3) are good solutions.
(4.1) and (4.4) should be rejected.
For (4.1), 12 sin x + 5 cos x 
						
						
						( 11.59951
						( 4
(	
 does not satisfy the equation 12 sin x + 5 cos x = 4, and should be rejected.
Similarly, for (4.4), 
 should be rejected.
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