Two angles of a triangle are `"tan"^(-1)(1)/(2)and "tan"^(-1)(1)/(3)`. What is the third angle ?

Options

A. `30^(@)`
B. `45^(@)`
C. `90^(@)`
D. `135^(@)`

Explanation

To find the third angle of a triangle when two angles are given as \( \tan^{-1} \left( \frac{1}{2} \right) \) and \( \tan^{-1} \left( \frac{1}{3} \right) \), we can follow these steps:

### Step 1: Identify the angles
Let:
- Angle A = \( \tan^{-1} \left( \frac{1}{2} \right) \)
- Angle B = \( \tan^{-1} \left( \frac{1}{3} \right) \)

### Step 2: Use the property of triangles
The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can express the third angle (Angle C) as:
\[ \text{Angle C} = 180^\circ - \text{Angle A} - \text{Angle B} \]

### Step 3: Calculate \( \tan(A + B) \)
To find Angle C, we first need to calculate \( A + B \) using the formula for the tangent of the sum of two angles:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Substituting \( \tan A = \frac{1}{2} \) and \( \tan B = \frac{1}{3} \):
\[ \tan(A + B) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left( \frac{1}{2} \cdot \frac{1}{3} \right)} \]

### Step 4: Simplify the expression
Calculating the numerator:
\[ \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
Calculating the denominator:
\[ 1 - \left( \frac{1}{2} \cdot \frac{1}{3} \right) = 1 - \frac{1}{6} = \frac{5}{6} \]
Thus,
\[ \tan(A + B) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]

### Step 5: Find \( A + B \)
Since \( \tan(A + B) = 1 \), we know:
\[ A + B = \tan^{-1}(1) = 45^\circ \]

### Step 6: Calculate Angle C
Now substituting back to find Angle C:
\[ \text{Angle C} = 180^\circ - (A + B) = 180^\circ - 45^\circ = 135^\circ \]

### Conclusion
Thus, the third angle of the triangle is:
\[ \text{Angle C} = 135^\circ \]