A variable trisriable triangle is inscribed in a circle ofus R. If the rate of change of a side is R times the rate of change of the opposite angle, then the opposite angle is

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Explanation

To solve the problem step by step, we will use the relationship between the side of a triangle inscribed in a circle and the angle opposite to it, along with the given rate of change conditions.

### Step 1: Understand the relationship between the side and the angle in a triangle inscribed in a circle
For a triangle inscribed in a circle of radius \( R \), the relationship between a side \( a \) and the opposite angle \( A \) is given by the formula:
\[ a = 2R \sin A \]

### Step 2: Differentiate the relationship with respect to time
We need to find the rate of change of the side \( a \) with respect to the angle \( A \). Differentiating both sides of the equation \( a = 2R \sin A \) with respect to time \( t \) gives:
\[ \frac{da}{dt} = 2R \cos A \frac{dA}{dt} \]

### Step 3: Use the given rate of change condition
According to the problem, the rate of change of the side \( a \) is \( R \) times the rate of change of the angle \( A \):
\[ \frac{da}{dt} = R \frac{dA}{dt} \]

### Step 4: Set the two expressions for \( \frac{da}{dt} \) equal to each other
From the differentiation, we have:
\[ 2R \cos A \frac{dA}{dt} = R \frac{dA}{dt} \]

### Step 5: Simplify the equation
Assuming \( \frac{dA}{dt} \neq 0 \), we can divide both sides by \( R \frac{dA}{dt} \):
\[ 2 \cos A = 1 \]

### Step 6: Solve for \( \cos A \)
From the equation \( 2 \cos A = 1 \), we can solve for \( \cos A \):
\[ \cos A = \frac{1}{2} \]

### Step 7: Find the angle \( A \)
The angle \( A \) for which \( \cos A = \frac{1}{2} \) is:
\[ A = \frac{\pi}{3} \quad \text{(or 60 degrees)} \]

### Conclusion
Thus, the opposite angle \( A \) is:
\[ A = \frac{\pi}{3} \]