Hey,
Given equation
Note y is displacement whose dimension is [L].
According to dimensional rule,
Dimension of y = Dimension of Asinbx.
We also know sin(x) or any trigonometric function is dimensionless.
Therefore, Dimension of y = Dimension of A
Implies, [y]=[A]. So, [A]=[L]A=[M^0L^1T^0]
Bx=[M^0L^0T^0]B=[M^0L^-1T^0]
Ct=[M^0L^0T^0]C=[M^0L^0T^-1]
D=[M^0L^0T^0] by multiplying (ABCD)
[M^0L^0T^0]×[M^0L^-1T^0]×[M^0L^0T^-1]×[M^0L^0T^0]=[M^0L^0T^-1]

Dimensional Formula for (ABCD) in the equation y=A sin (Bx Ct D)

The equation y=A sin (Bx Ct D) represents the displacement of an oscillating particle. The dimensional formula for the constants A, B, C, and D can be determined using the principle of homogeneity of dimensions. According to this principle, the dimensions of each term in an equation must be the same. Therefore, the dimensions of A, B, C, and D can be determined by analyzing the dimensions of each term in the equation.

Y

The dimension of y, which represents the displacement of the oscillating particle, is given by [L], where L represents the dimension of length.

A

The constant A represents the amplitude of the oscillation. Therefore, the dimension of A can be obtained by analyzing the dimension of y when x, t, and D are equal to zero. At these values, the equation reduces to y=A sin (0), which implies that y=A. Therefore, the dimension of A is also [L].

Bx

The term Bx represents the argument of the sine function and has a dimension of [1]. Therefore, the dimension of B can be obtained by analyzing the dimension of y when x, t, and D are equal to one. At these values, the equation reduces to y=A sin (B C D), which implies that y has a dimension of [L]. Therefore, the dimension of B is [T^-1].

Ct

The term Ct represents the argument of the sine function and has a dimension of [1]. Therefore, the dimension of C can be obtained by analyzing the dimension of y when x, t, and D are equal to one. At these values, the equation reduces to y=A sin (B C D), which implies that y has a dimension of [L]. Therefore, the dimension of C is [T^-1].

D

The constant D represents the phase of the oscillation and has a dimension of [1]. Therefore, the dimension of D is also [1].

Overall Dimensional Formula for (ABCD)

Based on the above analysis, the dimensional formula for (ABCD) in the equation y=A sin (Bx Ct D) can be written as [L] [T^-1] [T^-1] [1], which can be simplified as [L T^-2].