Find the number of integral values of k for which the equation 12cosx - 5 sinx = 2k+1 has a solution.

Explanation

Dividing both sides by `sqrt(12^(2)+5^(2))=13` we get
`12/13 cos x-5/13 sinx = (2k+1)/13`
`rArr cos(x + alpha) = (2k+1)/(13)`, where `tan alpha = 5/12`
In order that the given equation possesses a solution we must have,
`rArr -13 le 2k +1 le 13`
`rArr -14 le 2k le 12`
`rArr -7 le k le 6`
The number of integral values of k in `[-7,6]` is 14.