If `x sin theta = ( 5 sqrt3)/(2) and x cos theta = (5)/(2) ` then what is the value of x ?

Options

A. `sqrt3`
B. `1 //2`
C. `sqrt3//2`
D. `5`

Answer

To find the value of \( x \) given the equations \( x \sin \theta = \frac{5\sqrt{3}}{2} \) and \( x \cos \theta = \frac{5}{2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Write down the equations**:
\[ x \sin \theta = \frac{5\sqrt{3}}{2} \quad \text{(Equation 1)} \]
\[ x \cos \theta = \frac{5}{2} \quad \text{(Equation 2)} \]

2. **Divide Equation 1 by Equation 2**:
\[ \frac{x \sin \theta}{x \cos \theta} = \frac{\frac{5\sqrt{3}}{2}}{\frac{5}{2}} \]
This simplifies to:
\[ \frac{\sin \theta}{\cos \theta} = \frac{5\sqrt{3}/2}{5/2} \]

3. **Simplify the right side**:
\[ \frac{\sin \theta}{\cos \theta} = \frac{5\sqrt{3}}{2} \cdot \frac{2}{5} = \sqrt{3} \]
Thus, we have:
\[ \tan \theta = \sqrt{3} \]

4. **Find the angle \( \theta \)**:
The angle \( \theta \) where \( \tan \theta = \sqrt{3} \) is:
\[ \theta = 60^\circ \]

5. **Substitute \( \theta \) back into Equation 2** to find \( x \):
Using \( \cos 60^\circ = \frac{1}{2} \):
\[ x \cos 60^\circ = \frac{5}{2} \]
This becomes:
\[ x \cdot \frac{1}{2} = \frac{5}{2} \]

6. **Solve for \( x \)**:
Multiply both sides by 2:
\[ x = 5 \]

### Final Answer:
\[ x = 5 \]