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