If x `sintheta=(5sqrt(3))/(2)andxcostheta=(5)/(2)` , what is the value of x?

Options

A. `sqrt(3)`
B. `(1)/(2)`
C. `(sqrt(3))/(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 1: Write down the equations
We have:
1. \( x \sin \theta = \frac{5\sqrt{3}}{2} \) (Equation 1)
2. \( x \cos \theta = \frac{5}{2} \) (Equation 2)

### Step 2: Square both equations
Square Equation 1:
\[ (x \sin \theta)^2 = \left(\frac{5\sqrt{3}}{2}\right)^2 \]
This gives:
\[ x^2 \sin^2 \theta = \frac{25 \cdot 3}{4} = \frac{75}{4} \quad \text{(Equation 3)} \]

Square Equation 2:
\[ (x \cos \theta)^2 = \left(\frac{5}{2}\right)^2 \]
This gives:
\[ x^2 \cos^2 \theta = \frac{25}{4} \quad \text{(Equation 4)} \]

### Step 3: Add the two squared equations
Now, we add Equation 3 and Equation 4:
\[ x^2 \sin^2 \theta + x^2 \cos^2 \theta = \frac{75}{4} + \frac{25}{4} \]
This simplifies to:
\[ x^2 (\sin^2 \theta + \cos^2 \theta) = \frac{100}{4} = 25 \]

### Step 4: Use the Pythagorean identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can substitute:
\[ x^2 \cdot 1 = 25 \]
This simplifies to:
\[ x^2 = 25 \]

### Step 5: Take the square root
Taking the square root of both sides gives:
\[ x = \sqrt{25} = 5 \]

### Final Answer
The value of \( x \) is \( 5 \).

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