A physical quantity x depends on qualities y and z as...

A physical quantity $x$ depends on qualities $y$ and $z$ as follows: $x=A y+B \tan C z$, where $A, B$ and $C$ are constants. Which of the following do not have the same dimensions?
(A) $x$ and $B$
(B) $C$ and $z^{-1}$
(C) $y$ and $B / A$
(D) $x$ and $A$

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Key Concepts

Dimensional Analysis

Dimensional analysis is the process of checking the consistency of the dimensions on both sides of an equation. It ensures that physical equations make sense by requiring all terms that are added or equated to each other to have the same dimensions. This method is vital for verifying correct relationships between physical quantities and is widely used in physics and engineering.

Dimensionless Arguments in Trigonometric Functions

Trigonometric functions require their arguments to be dimensionless. This means when you apply a trigonometric function (such as tangent), its argument must be a pure number without any physical units. When variables with dimensions are involved within trigonometric functions, they are typically multiplied by constants that have appropriate dimensions to cancel out the units, ensuring the argument is dimensionless.

Physical Constants and Their Dimensions

Physical constants in equations play a role in balancing the dimensions of the variables involved. These constants can carry dimensions that are necessary to make the overall equation dimensionally consistent. Their dimensions are determined by the requirement that, when combined with the variables, the resulting expressions conform to the rules of dimensional homogeneity.

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