In `DeltaABC, if A = 30^@,b= 8,a=6, B = sin^-1 x`, then `x=`

Options

A. `1//2`
B. `1//3`
C. `2//3`
D. 1

Answer

To solve the problem, we will use the sine rule in triangle ABC. The sine rule states that:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

Where:
- \( A = 30^\circ \)
- \( a = 6 \)
- \( b = 8 \)
- \( B = \sin^{-1}(x) \)

### Step-by-step Solution:

1. **Write the Sine Rule**:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

2. **Substitute the known values**:
\[ \frac{6}{\sin(30^\circ)} = \frac{8}{\sin(\sin^{-1}(x))} \]

3. **Calculate \(\sin(30^\circ)\)**:
\[ \sin(30^\circ) = \frac{1}{2} \]

4. **Substitute \(\sin(30^\circ)\) into the equation**:
\[ \frac{6}{\frac{1}{2}} = \frac{8}{x} \]

5. **Simplify the left side**:
\[ 6 \times 2 = \frac{8}{x} \]
\[ 12 = \frac{8}{x} \]

6. **Cross-multiply to solve for \(x\)**:
\[ 12x = 8 \]

7. **Divide both sides by 12**:
\[ x = \frac{8}{12} \]

8. **Simplify the fraction**:
\[ x = \frac{2}{3} \]

### Final Answer:
\[ x = \frac{2}{3} \]