The difference of two angles is `1^(@)` , the circular measure of their sum is `1` . What is the smaller angle in circular measure ?

Options

A. `[(180)/(pi)-1]`
B. `[1-(pi)/(180)]`
C. `(1)/(2)[1-(pi)/(180)]`
D. `(1)/(2)[(180)/(pi)-1]`

Explanation

To solve the problem step by step, we will define the two angles and set up equations based on the information provided.

### Step 1: Define the Angles
Let the two angles be \( \alpha \) and \( \beta \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The difference of the two angles is \( 1^\circ \):
\[ \alpha - \beta = 1^\circ \]
2. The circular measure of their sum is \( 1 \) (in radians):
\[ \alpha + \beta = 1 \text{ radian} \]

### Step 3: Convert Degrees to Radians
Since \( 1^\circ \) in radians is given by:
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]
We can rewrite the first equation:
\[ \alpha - \beta = \frac{\pi}{180} \]

### Step 4: Solve the System of Equations
Now we have the following system of equations:
1. \( \alpha - \beta = \frac{\pi}{180} \) (Equation 1)
2. \( \alpha + \beta = 1 \) (Equation 2)

To solve for \( \alpha \), we can add both equations:
\[ (\alpha - \beta) + (\alpha + \beta) = \frac{\pi}{180} + 1 \]
This simplifies to:
\[ 2\alpha = 1 + \frac{\pi}{180} \]
Thus, we can solve for \( \alpha \):
\[ \alpha = \frac{1 + \frac{\pi}{180}}{2} \]

### Step 5: Solve for \( \beta \)
Now, we can find \( \beta \) using Equation 2:
\[ \alpha + \beta = 1 \]
Substituting \( \alpha \):
\[ \frac{1 + \frac{\pi}{180}}{2} + \beta = 1 \]
Rearranging gives:
\[ \beta = 1 - \frac{1 + \frac{\pi}{180}}{2} \]
This simplifies to:
\[ \beta = \frac{2 - (1 + \frac{\pi}{180})}{2} = \frac{1 - \frac{\pi}{180}}{2} \]

### Step 6: Determine the Smaller Angle
Now we have both angles:
- \( \alpha = \frac{1 + \frac{\pi}{180}}{2} \)
- \( \beta = \frac{1 - \frac{\pi}{180}}{2} \)

Since \( \frac{\pi}{180} > 0 \), it follows that \( \beta < \alpha \). Therefore, the smaller angle in circular measure is:
\[ \beta = \frac{1 - \frac{\pi}{180}}{2} \]

### Final Answer
The smaller angle in circular measure is:
\[ \beta = \frac{1 - \frac{\pi}{180}}{2} \]