Let `sin(A+B)=1andsin(A-B)=(1)/(2)` where A,B `in[0,(pi)/(2)]`. What is the value of A ?

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `(pi)/(4)`
D. `(pi)/(8)`

Explanation

To solve the problem, we start with the given equations:

1. \( \sin(A + B) = 1 \)
2. \( \sin(A - B) = \frac{1}{2} \)

### Step 1: Analyze the first equation
From the first equation, \( \sin(A + B) = 1 \), we know that sine reaches its maximum value of 1 at \( 90^\circ \) (or \( \frac{\pi}{2} \) radians). Therefore, we can set up the equation:

\[ A + B = 90^\circ \quad \text{(or)} \quad A + B = \frac{\pi}{2} \]

### Step 2: Analyze the second equation
From the second equation, \( \sin(A - B) = \frac{1}{2} \). We know that sine equals \( \frac{1}{2} \) at \( 30^\circ \) (or \( \frac{\pi}{6} \) radians). Thus, we can set up the equation:

\[ A - B = 30^\circ \quad \text{(or)} \quad A - B = \frac{\pi}{6} \]

### Step 3: Set up the system of equations
Now we have a system of two equations:

1. \( A + B = 90^\circ \)
2. \( A - B = 30^\circ \)

### Step 4: Solve the system of equations
To find the value of \( A \), we can add the two equations:

\[ (A + B) + (A - B) = 90^\circ + 30^\circ \]

This simplifies to:

\[ 2A = 120^\circ \]

Dividing both sides by 2 gives:

\[ A = 60^\circ \]

### Step 5: Convert to radians
Since the question asks for the value of \( A \) in radians, we convert \( 60^\circ \) to radians:

\[ A = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \]

### Final Answer
Thus, the value of \( A \) is:

\[ \boxed{\frac{\pi}{3}} \]