Correct: y = a sin(2πt/T)
Dimension of y = M0 L1 T0
Dimension of a = M0 L1 T0
Dimension of L.H.S = Dimension of R.H.S
Hence, the given formula is dimensionally correct.
Incorrect y = a sin vt
Dimension of y = M0 L1 T0
Dimension of a = M0 L1 T0
Dimension of vt = M0 L1 T-1 x M0 L0 T1 = M0 L1 T0
But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the given formula is dimensionally incorrect.
Incorrect y = (a/T) sin(t/a)
Dimension of y = M0 L1 T0
Dimension of a/T = M0 L1 T-1
Dimension of t/a = M0 L-1 T1
But the argument of the trigonometric function must be dimensionless, which is not so in the given case. Hence, the formula is dimensionally incorrect.
Correct
Since the argument of the trigonometric function must be dimensionless (which is true in the given case), the dimensions of y and a are the same. Hence, the given formula is dimensionally correct.