The area of a triangleis computed using the formula `S=1/2 bc sin A`. If the relative errors made in measuring b,c and calculating S are respectively 0.02,0.01 and 0.13 the approximate error in A when `A=pi//6`, is

Options

A. 0.05 radians
B. 0.01 radians
C. 0.05 degree
D. 0.01 degree

Explanation

To find the approximate error in angle \( A \) when the area \( S \) of a triangle is given by the formula \( S = \frac{1}{2}bc \sin A \), we will follow these steps:

### Step 1: Write down the formula for the area of the triangle
The area \( S \) is given by:
\[ S = \frac{1}{2}bc \sin A \]

### Step 2: Differentiate the area with respect to its variables
We will differentiate \( S \) with respect to \( b \), \( c \), and \( A \) using the product and chain rules:
\[ dS = \frac{1}{2}(c \sin A \, db + b \sin A \, dc + bc \cos A \, dA) \]

### Step 3: Divide by \( S \) to find the relative error
To find the relative error, we divide both sides by \( S \):
\[ \frac{dS}{S} = \frac{1}{2} \left( \frac{c \sin A}{S} \, db + \frac{b \sin A}{S} \, dc + \frac{bc \cos A}{S} \, dA \right) \]

### Step 4: Substitute \( S \) into the equation
Since \( S = \frac{1}{2}bc \sin A \), we can substitute this into the equation:
\[ \frac{dS}{S} = \frac{c \sin A}{bc \sin A} \, db + \frac{b \sin A}{bc \sin A} \, dc + \frac{bc \cos A}{bc \sin A} \, dA \]
This simplifies to:
\[ \frac{dS}{S} = \frac{1}{b} \, db + \frac{1}{c} \, dc + \cot A \, dA \]

### Step 5: Substitute the known relative errors
Given the relative errors:
- \( \frac{db}{b} = 0.02 \)
- \( \frac{dc}{c} = 0.01 \)
- \( \frac{dS}{S} = 0.13 \)

Substituting these values into the equation:
\[ 0.13 = 0.02 + 0.01 + \cot A \, dA \]

### Step 6: Solve for \( dA \)
Rearranging gives:
\[ \cot A \, dA = 0.13 - 0.02 - 0.01 = 0.10 \]
Thus,
\[ dA = \frac{0.10}{\cot A} \]

### Step 7: Calculate \( \cot A \) when \( A = \frac{\pi}{6} \)
We know:
\[ \cot \left(\frac{\pi}{6}\right) = \frac{1}{\tan \left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Step 8: Substitute \( \cot A \) back into the equation for \( dA \)
Now substituting \( \cot A \):
\[ dA = \frac{0.10}{\sqrt{3}} \]

### Step 9: Calculate the numerical value
Calculating \( dA \):
\[ dA \approx \frac{0.10}{1.732} \approx 0.0577 \text{ radians} \]

### Final Answer
The approximate error in \( A \) when \( A = \frac{\pi}{6} \) is approximately:
\[ dA \approx 0.0577 \text{ radians} \approx 0.05 \text{ radians} \]