One of the angles of a triangle is `1//2` radian and the other is `99^(@)` . What is the third angle in radian measure ?

Options

A. `(9pi-10)/(pi)`
B. `(90pi-100)/(7pi)`
C. `(90pi-10)/(pi)`
D. None of these

Explanation

To find the third angle of the triangle given that one angle is \( \frac{1}{2} \) radian and the other is \( 99^\circ \), we can follow these steps:

### Step 1: Convert \( 99^\circ \) to Radians
To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \).

\[ 99^\circ = 99 \times \frac{\pi}{180} = \frac{99\pi}{180} \]

### Step 2: Simplify the Radian Measure
Now, we simplify \( \frac{99\pi}{180} \):

\[ \frac{99\pi}{180} = \frac{11\pi}{20} \]

### Step 3: Use the Triangle Angle Sum Property
The sum of the angles in a triangle is \( \pi \) radians. Let the angles be \( A \), \( B \), and \( C \). We know:

\[ A + B + C = \pi \]

Substituting the known values:

\[ \frac{1}{2} + \frac{11\pi}{20} + C = \pi \]

### Step 4: Solve for \( C \)
Rearranging the equation to solve for \( C \):

\[ C = \pi - \left(\frac{1}{2} + \frac{11\pi}{20}\right) \]

### Step 5: Combine the Terms
First, we need a common denominator to combine \( \frac{1}{2} \) and \( \frac{11\pi}{20} \). The common denominator is 20.

\[ \frac{1}{2} = \frac{10}{20} \]

Now substituting:

\[ C = \pi - \left(\frac{10}{20} + \frac{11\pi}{20}\right) = \pi - \frac{10 + 11\pi}{20} \]

### Step 6: Simplify Further
Now, we can express \( \pi \) as \( \frac{20\pi}{20} \):

\[ C = \frac{20\pi}{20} - \frac{10 + 11\pi}{20} = \frac{20\pi - 10 - 11\pi}{20} \]

### Step 7: Final Calculation
Simplifying the numerator:

\[ C = \frac{20\pi - 11\pi - 10}{20} = \frac{9\pi - 10}{20} \]

### Final Answer
Thus, the third angle \( C \) in radian measure is:

\[ C = \frac{9\pi - 10}{20} \]

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