If two angles of a triangle are `tan^(-1) ""(1)/(2) and tan^(-1)"" (1)/(3)` . Then the third angle is

Options

A. `(pi)/(4)`
B. `(5 pi)/(4)`
C. `(pi)/(2)`
D. `(3 pi)/(4)`

Answer

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

### Step 1: Understand the sum of angles in a triangle
The sum of all angles in a triangle is \( 180^\circ \) or \( \pi \) radians. Let the third angle be \( x \).

### Step 2: Set up the equation
We can write the equation for the sum of angles:
\[ \tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{3} \right) + x = \pi \]
From this, we can express \( x \) as:
\[ x = \pi - \left( \tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{3} \right) \right) \]

### Step 3: Use the formula for the sum of arctangents
We can use the formula for the sum of two arctangents:
\[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \]
Here, let \( a = \frac{1}{2} \) and \( b = \frac{1}{3} \).

### Step 4: Calculate \( a + b \) and \( 1 - ab \)
Calculating \( a + b \):
\[ a + b = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
Calculating \( 1 - ab \):
\[ ab = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \quad \Rightarrow \quad 1 - ab = 1 - \frac{1}{6} = \frac{5}{6} \]

### Step 5: Substitute into the formula
Now we can substitute into the formula:
\[ \tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{3} \right) = \tan^{-1} \left( \frac{\frac{5}{6}}{\frac{5}{6}} \right) = \tan^{-1}(1) \]

### Step 6: Find the angle
Since \( \tan^{-1}(1) = \frac{\pi}{4} \), we have:
\[ \tan^{-1} \left( \frac{1}{2} \right) + \tan^{-1} \left( \frac{1}{3} \right) = \frac{\pi}{4} \]

### Step 7: Substitute back to find \( x \)
Now substitute back to find \( x \):
\[ x = \pi - \frac{\pi}{4} = \frac{4\pi}{4} - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Conclusion
Thus, the third angle \( x \) is:
\[ \boxed{\frac{3\pi}{4}} \]