Find the general solution of each of the following equations:

(i) cos 3x = cos 2x


(ii) cos 5x = sin 3x


(iii) cos mx = sin nx



To Find: General solution.


(i) Given: cos 3x = cos 2x cos 3x - cos 2x = 0 -2sin sin = 0


[NOTE: cos C – cos D = -2sin sin ]


So, sin = 0 or sin= 0


Formula used: sin = 0 = n , n I


= n or = m where n, m I


x = 2 n/5 or x = 2m where n, m I


So general solution is x = 2 n/5 or x = 2m where n, m I


(ii) Given: cos 5x = sin 3x cos 5x = cos


Formula used: cos = cos = 2n , n I


By using the above formula, we have


5x = 2n or 5x = 2n


8x = 2n or 2x = 2n


x = or x = n where n I


So general solution is x = or x = n where n I


(iii) Given: cos mx = sin nx cos mx = cos


Formula used: cos = cos = 2k , k I


By using the above formula, we have


mx = 2k or 5x = 2k


(m+n)x = 2k or (m-n)x = 2k


x = or x = where k I


x = or x = where k I


So the general solution is x = or x = where k I


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