If r `sintheta=(7)/(2)andrcostheta=(7sqrt(3))/(2)` then the value of `theta` is :

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

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

1. \( r \sin \theta = \frac{7}{2} \)
2. \( r \cos \theta = \frac{7\sqrt{3}}{2} \)

### Step 1: Express \( r \) in terms of \( \theta \)

From the first equation, we can express \( r \) as:
\[ r = \frac{7}{2 \sin \theta} \]

From the second equation, we can express \( r \) as:
\[ r = \frac{7\sqrt{3}}{2 \cos \theta} \]

### Step 2: Set the two expressions for \( r \) equal to each other

Since both expressions represent \( r \), we can set them equal:
\[ \frac{7}{2 \sin \theta} = \frac{7\sqrt{3}}{2 \cos \theta} \]

### Step 3: Simplify the equation

We can cancel \( 7 \) and \( 2 \) from both sides:
\[ \frac{1}{\sin \theta} = \frac{\sqrt{3}}{\cos \theta} \]

### Step 4: Cross-multiply to eliminate the fractions

Cross-multiplying gives us:
\[ \cos \theta = \sqrt{3} \sin \theta \]

### Step 5: Divide both sides by \( \cos \theta \)

Dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ 1 = \sqrt{3} \tan \theta \]

### Step 6: Solve for \( \tan \theta \)

Rearranging gives:
\[ \tan \theta = \frac{1}{\sqrt{3}} \]

### Step 7: Find the angle \( \theta \)

We know that:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus, we find:
\[ \theta = 30^\circ \]

### Final Answer

The value of \( \theta \) is \( 30^\circ \).

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