Let `theta` be a positive angle. Is the number of degrees in is divided by the number of radians in `theta`, then an irrational number `(180)/(pi)` results. If the number of degrees in 0 is multiplied by the number of radians in `theta`. then an irrational number `(125pi)/(9)` results. The angle `theta` must be equal to:

Options

A. `30^@`
B. `45^@`
C. `50^@`
D. `60^@`

Answer

To solve the problem, we will follow the steps outlined in the video transcript.

### Step 1: Define the Variables
Let:
- \( \theta_d \) = angle in degrees
- \( \theta_r \) = angle in radians

### Step 2: Set Up the First Equation
According to the problem, when the number of degrees is divided by the number of radians, we have:
\[ \frac{\theta_d}{\theta_r} = \frac{180}{\pi} \]
From this, we can express \( \theta_r \) in terms of \( \theta_d \):
\[ \theta_r = \frac{\pi}{180} \theta_d \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that when the number of degrees is multiplied by the number of radians, we have:
\[ \theta_d \cdot \theta_r = \frac{125\pi}{9} \]
Substituting Equation 1 into this equation gives:
\[ \theta_d \cdot \left(\frac{\pi}{180} \theta_d\right) = \frac{125\pi}{9} \]
This simplifies to:
\[ \frac{\pi}{180} \theta_d^2 = \frac{125\pi}{9} \]

### Step 4: Eliminate \( \pi \) and Solve for \( \theta_d^2 \)
We can cancel \( \pi \) from both sides:
\[ \frac{1}{180} \theta_d^2 = \frac{125}{9} \]
Now, multiply both sides by 180:
\[ \theta_d^2 = 180 \cdot \frac{125}{9} \]
Calculating the right side:
\[ \theta_d^2 = \frac{180 \cdot 125}{9} = \frac{22500}{9} = 2500 \]

### Step 5: Find \( \theta_d \)
Taking the square root of both sides:
\[ \theta_d = \sqrt{2500} = 50 \]
Since \( \theta \) is a positive angle, we take the positive value:
\[ \theta_d = 50 \text{ degrees} \]

### Step 6: Conclusion
Thus, the angle \( \theta \) must be equal to \( 50 \) degrees.