If `tantheta=(1)/(2)andtanphi=(1)/(3)`, then `theta+phi=?`

Options

A. `(pi)/(6)`
B. 0
C. `(pi)/(4)`
D. `pi`

Answer

To find the value of \(\theta + \phi\) given that \(\tan \theta = \frac{1}{2}\) and \(\tan \phi = \frac{1}{3}\), we can use the formula for the tangent of the sum of two angles:

\[ \tan(\theta + \phi) = \frac{\tan \theta + \tan \phi}{1 - \tan \theta \tan \phi} \]

### Step 1: Substitute the values of \(\tan \theta\) and \(\tan \phi\)

We know:
\[ \tan \theta = \frac{1}{2} \quad \text{and} \quad \tan \phi = \frac{1}{3} \]

Now, substituting these values into the formula:

\[ \tan(\theta + \phi) = \frac{\frac{1}{2} + \frac{1}{3}}{1 - \left(\frac{1}{2} \cdot \frac{1}{3}\right)} \]

### Step 2: Calculate the numerator

To add \(\frac{1}{2}\) and \(\frac{1}{3}\), we need a common denominator. The least common multiple of 2 and 3 is 6.

\[ \tan \theta + \tan \phi = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

### Step 3: Calculate the denominator

Now, we calculate \(1 - \tan \theta \tan \phi\):

\[ \tan \theta \tan \phi = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6} \]

Thus, the denominator becomes:

\[ 1 - \tan \theta \tan \phi = 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

### Step 4: Substitute back into the formula

Now we can substitute the values back into the formula:

\[ \tan(\theta + \phi) = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \]

### Step 5: Find \(\theta + \phi\)

Since \(\tan(\theta + \phi) = 1\), we know that:

\[ \theta + \phi = 45^\circ \quad \text{(or } \frac{\pi}{4} \text{ radians)} \]

### Final Answer

Thus, the value of \(\theta + \phi\) is:

\[ \theta + \phi = 45^\circ \]

---