The percentage error in measurement of a physical quantity [m given by `m=pi tan theta`] is minimum when
(Assume that error in `theta` remain constant)

Options

A. `theta=45^(@)`
B. `theta=90^(@)`
C. `theta=60^(@)`
D. `theta=30^(@)`

Answer

To find the angle \( \theta \) at which the percentage error in the measurement of the physical quantity \( m \) given by \( m = \pi \tan \theta \) is minimized, we can follow these steps:

### Step 1: Understand the relationship
The physical quantity is given by:
\[ m = \pi \tan \theta \]
We need to find the percentage error in \( m \).

### Step 2: Differentiate \( m \)
To find the change in \( m \) with respect to \( \theta \), we differentiate \( m \):
\[ dm = \pi \sec^2 \theta \, d\theta \]
where \( \sec^2 \theta \) is the derivative of \( \tan \theta \).

### Step 3: Express the relative error
The relative error in \( m \) can be expressed as:
\[ \frac{dm}{m} = \frac{\pi \sec^2 \theta \, d\theta}{\pi \tan \theta} \]
This simplifies to:
\[ \frac{dm}{m} = \frac{\sec^2 \theta}{\tan \theta} \, d\theta \]

### Step 4: Simplify the expression
Using the definitions of \( \sec \) and \( \tan \):
\[ \sec^2 \theta = \frac{1}{\cos^2 \theta}, \quad \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus, we can rewrite the expression:
\[ \frac{dm}{m} = \frac{1}{\cos^2 \theta} \cdot \frac{\cos \theta}{\sin \theta} \, d\theta = \frac{1}{\sin \theta \cos \theta} \, d\theta \]

### Step 5: Use the double angle identity
Recognizing that:
\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]
We can rewrite the expression for relative error:
\[ \frac{dm}{m} = \frac{2 \, d\theta}{\sin(2\theta)} \]

### Step 6: Analyze the percentage error
The percentage error is inversely proportional to \( \sin(2\theta) \). Therefore, to minimize the percentage error, we need to maximize \( \sin(2\theta) \).

### Step 7: Find the maximum value of \( \sin(2\theta) \)
The maximum value of \( \sin(2\theta) \) is 1, which occurs when:
\[ 2\theta = 90^\circ \quad \Rightarrow \quad \theta = 45^\circ \]

### Conclusion
Thus, the percentage error in the measurement of \( m \) is minimized when:
\[ \theta = 45^\circ \]

### Final Answer
The answer is:
**Option A: \( \theta = 45^\circ \)**