$A$ book with many printing errors contains four different formulas for the displacement $y$ of a particle undergoing a certain periodic motion:
$(a) \; y = a \sin \left(\frac{2 \pi t}{T}\right)$
$(b) \; y = a \sin v t$
$(c) \; y = \left(\frac{a}{T}\right) \sin \frac{t}{a}$
$(d) \; y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
($a =$ maximum displacement of the particle,$v =$ speed of the particle,$T =$ time-period of motion). Rule out the wrong formulas on dimensional grounds.

(B, C) Correct: $y = a \sin \left(\frac{2 \pi t}{T}\right)$
Dimension of $y = [L]$. Dimension of $a = [L]$. The argument of $\sin$ is $\frac{2 \pi t}{T}$,which is dimensionless. Thus,the formula is dimensionally correct.
$(b)$ Incorrect: $y = a \sin v t$
Dimension of $v t = [LT^{-1}] \times [T] = [L]$. The argument of the trigonometric function must be dimensionless,but here it has dimensions of length. Thus,it is dimensionally incorrect.
$(c)$ Incorrect: $y = \left(\frac{a}{T}\right) \sin \left(\frac{t}{a}\right)$
Dimension of $\frac{a}{T} = [LT^{-1}] \neq [L]$. Also,the argument $\frac{t}{a} = [TL^{-1}]$ is not dimensionless. Thus,it is dimensionally incorrect.
$(d)$ Correct: $y = (a \sqrt{2}) \left(\sin \frac{2 \pi t}{T} + \cos \frac{2 \pi t}{T}\right)$
Dimension of $y = [L]$. Dimension of $a = [L]$. The argument $\frac{2 \pi t}{T}$ is dimensionless. Thus,the formula is dimensionally correct.