Sir

 Why do we use the cosine equation for some SHM problems and sine equation used for some other problems?

Can we use both of them interchangeably for the same problem ? If yes then please give an example problem for it. 

 

Depending on the question, sine or cosine function is chosen.

If the question states that a particle starts at maximum displacement position and then performs SHM, then it is necessary that you take cosine function. This is because at t = 0 only cos 0 will give maximum value and hence maximum amplitude.

 

If the question says that the particle starts from mean position, then it is necessary that we use the sine function. This is because at t = 0 only sin 0 will give minimum value and hence minimum amplitude.

 

Now, if the question says that at t - 0 the particle is at some displacement (not maximum or minimum) then one can take sin or cos interchangeably. For example question 14.7 inn textbook gives that particle is at some position x = 1 cm at t = 0. In this case sin or cos will not make a difference. The question itself asks to change the function from cos to sin. What will however change is the amplitude and phase because of changing to sin. Overall results will not be affected at all.