The value of x satisfying the equation `sinx+(1)/(sinx)=(7)/(2sqrt(3))` is given by -

Options

A. `10^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Explanation

To solve the equation \( \sin x + \frac{1}{\sin x} = \frac{7}{2\sqrt{3}} \), we can follow these steps:

### Step 1: Multiply through by \( \sin x \)
To eliminate the fraction, we multiply both sides of the equation by \( \sin x \):
\[ \sin^2 x + 1 = \frac{7}{2\sqrt{3}} \sin x \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ \sin^2 x - \frac{7}{2\sqrt{3}} \sin x + 1 = 0 \]

### Step 3: Let \( y = \sin x \)
We can substitute \( y \) for \( \sin x \) to simplify our notation:
\[ y^2 - \frac{7}{2\sqrt{3}} y + 1 = 0 \]

### Step 4: Use the quadratic formula
Now, we can apply the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -\frac{7}{2\sqrt{3}} \), and \( c = 1 \):
\[ y = \frac{\frac{7}{2\sqrt{3}} \pm \sqrt{\left(-\frac{7}{2\sqrt{3}}\right)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]

### Step 5: Calculate the discriminant
Calculating the discriminant:
\[ \left(-\frac{7}{2\sqrt{3}}\right)^2 = \frac{49}{12} \]
\[ 4ac = 4 \cdot 1 \cdot 1 = 4 \]
Now, we need to express 4 with a common denominator:
\[ 4 = \frac{48}{12} \]
Thus, the discriminant becomes:
\[ \frac{49}{12} - \frac{48}{12} = \frac{1}{12} \]

### Step 6: Substitute back into the quadratic formula
Now substituting back into the quadratic formula:
\[ y = \frac{\frac{7}{2\sqrt{3}} \pm \sqrt{\frac{1}{12}}}{2} \]
Calculating \( \sqrt{\frac{1}{12}} = \frac{1}{2\sqrt{3}} \):
\[ y = \frac{\frac{7}{2\sqrt{3}} \pm \frac{1}{2\sqrt{3}}}{2} \]
\[ y = \frac{7 \pm 1}{4\sqrt{3}} \]

### Step 7: Solve for \( y \)
This gives us two possible values for \( y \):
\[ y_1 = \frac{8}{4\sqrt{3}} = \frac{2}{\sqrt{3}}, \quad y_2 = \frac{6}{4\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{\sqrt{3}}{2} \]

### Step 8: Find \( x \)
Now we find \( x \) for both values of \( y \):
1. For \( y_1 = \frac{2}{\sqrt{3}} \):
\[ \sin x = \frac{2}{\sqrt{3}} \quad \text{(not valid since } \sin x \leq 1\text{)} \]

2. For \( y_2 = \frac{\sqrt{3}}{2} \):
\[ \sin x = \frac{\sqrt{3}}{2} \implies x = 60^\circ \text{ or } x = 120^\circ \]

### Final Answer
The value of \( x \) satisfying the equation is \( 60^\circ \) or \( 120^\circ \).