Question 1187250: solve cos(3x) = cos(0.5x) for 0 <= x <= 180 degrees
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
  
Write both angles in terms of their average.
Their average is   
 and 
Substituting in original equation,
 
 
 
 
 ; 
 ;  ;
 and  so the above is
 ; 
  ; 
 ; 
But since  ,
 ,  ,
 ,  ,
 , ,
 ,  ,
Since n is an integer, n is either 0 or 1.
So the solutions are
 ; 
that is, x=0, and
 ; 
 ; 
So there are 3 solutions.
 ,  , and 
Edwin
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
I will get the same resulting answer, as Edwin, but will make my analysis differently.
To simplify my writing, I will introduce the angle y =  and
will look for angles y such that cos(6y) = cos(y) and 0 <= y <= 90 degrees.
Since cos(6y) = cos(y), it implies one of two possibilities:
(1) EITHER 6y = y + 360n degrees
(2) OR 6y = -y + 360n degrees.
From (1), I have 6y-y = 360n; 5y = 360n; y = 0, 360/5 = 72, (360/5)*2, . . .
Taking into account the restriction on the range, the only possible solutions are 0° and 72°.
From (2), I have 6y+y = 360n; 7y = 360n; y = 0, 360/7 = 51.43 degrees, (360/7)*2, . . .
Taking into account the restriction on the range, the only possible solutions are 0° and 360/7 degrees.
It gives the ANSWER : the only solutions for x are 0 degrees, 2*(360/5) = 144 degrees and 2*(360/7) degrees.
Solved.
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