If `Z = (A sin theta + B cos theta)/(A + B)`, then

Options

A. the dimensions of Z and A are the same
B. the dimensions of Z and B are the same
C. Z is dimensionless quantity
D. none of these

Answer

To solve the problem, we need to analyze the expression given:

\[ Z = \frac{A \sin \theta + B \cos \theta}{A + B} \]

We want to determine the relationship between the dimensions of \( A \), \( B \), and \( Z \).

### Step 1: Understand the components of the expression

The expression consists of two parts in the numerator: \( A \sin \theta \) and \( B \cos \theta \). The denominator is \( A + B \).

### Step 2: Analyze the dimensions of \( A \) and \( B \)

For the addition in the denominator \( A + B \) to be valid, the dimensions of \( A \) and \( B \) must be the same. This means:

\[ [A] = [B] \]

where \( [X] \) denotes the dimension of quantity \( X \).

### Step 3: Analyze the trigonometric functions

The sine and cosine functions are dimensionless. Therefore, we can say:

\[ [A \sin \theta] = [A] \]
\[ [B \cos \theta] = [B] \]

This means that the dimensions of \( A \sin \theta \) and \( B \cos \theta \) are simply the dimensions of \( A \) and \( B \), respectively.

### Step 4: Combine the dimensions in the numerator

Now, in the numerator, we have:

\[ [A \sin \theta + B \cos \theta] = [A] \]

Since both terms have the same dimension, we can conclude that:

\[ [A \sin \theta + B \cos \theta] = [A] \]

### Step 5: Analyze the denominator

The denominator \( A + B \) also has the dimension:

\[ [A + B] = [A] \]

### Step 6: Combine the dimensions of the entire expression

Now, we can substitute back into the expression for \( Z \):

\[ Z = \frac{[A]}{[A]} \]

### Step 7: Determine the dimensions of \( Z \)

Since both the numerator and the denominator have the same dimension, they cancel out:

\[ Z = \frac{[A]}{[A]} = 1 \]

This means that \( Z \) is dimensionless.

### Conclusion

Thus, we conclude that \( Z \) is a dimensionless quantity.

### Final Answer

The correct option is that \( Z \) is a dimensionless quantity.

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