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

### Step 1: Understand the relationship
We are given the equation \( m = \pi \tan \theta \). The percentage error in \( m \) will depend on the error in \( \theta \).

### Step 2: Differentiate the equation
To find the relationship between the errors, we differentiate \( m \) with respect to \( \theta \):
\[ \frac{dm}{d\theta} = \pi \sec^2 \theta \]

### Step 3: Express the error in terms of \( dm \) and \( d\theta \)
Using the differentiation result, we can express \( dm \) as:
\[ dm = \pi \sec^2 \theta \, d\theta \]

### Step 4: Find 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:
\[ \frac{dm}{m} = \frac{\sec^2 \theta \, d\theta}{\tan \theta} \]

### Step 5: Rewrite the expression
Using the identities \( \sec^2 \theta = 1 + \tan^2 \theta \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), we can rewrite the expression:
\[ \frac{dm}{m} = \frac{d\theta}{\sin \theta \cos \theta} \]
This can be further simplified to:
\[ \frac{dm}{m} = \frac{2 d\theta}{\sin 2\theta} \]

### Step 6: Analyze the function
To minimize the percentage error, we need to maximize the function \( \sin 2\theta \) since \( d\theta \) is a constant small angle. The maximum value of \( \sin 2\theta \) is 1.

### Step 7: Find the angle
The maximum value of \( \sin 2\theta = 1 \) occurs when:
\[ 2\theta = \frac{\pi}{2} \quad \Rightarrow \quad \theta = \frac{\pi}{4} \quad \Rightarrow \quad \theta = 45^\circ \]

### Conclusion
Thus, the percentage error in the measurement of the physical quantity \( m \) is minimized when \( \theta = 45^\circ \).

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