If tan(A) = 1/2 and tan(B) = 1/3, find the value of tan(A + B).

Steps to Solve

1. Write down the formula

The formula for the tangent of the sum of two angles is: $$\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$

2. Substitute the given values

We are given that $\tan(A) = \frac{2}{3}$ and $\tan(B) = \frac{1}{2}$. Substituting these values into the formula, we get:

$$\tan(A + B) = \frac{\frac{2}{3} + \frac{1}{2}}{1 - \frac{2}{3} \cdot \frac{1}{2}}$$

3. Simplify the numerator

To simplify the numerator, $\frac{2}{3} + \frac{1}{2}$, we find a common denominator, which is 6. So,

$$\frac{2}{3} + \frac{1}{2} = \frac{2 \cdot 2}{3 \cdot 2} + \frac{1 \cdot 3}{2 \cdot 3} = \frac{4}{6} + \frac{3}{6} = \frac{7}{6}$$

4. Simplify the denominator

To simplify the denominator, $1 - \frac{2}{3} \cdot \frac{1}{2}$, we first multiply the fractions:

$$\frac{2}{3} \cdot \frac{1}{2} = \frac{2 \cdot 1}{3 \cdot 2} = \frac{2}{6} = \frac{1}{3}$$

Then, subtract this result from 1:

$$1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$$

5. Divide the numerator by the denominator

Now we have:

$$\tan(A + B) = \frac{\frac{7}{6}}{\frac{2}{3}}$$

To divide fractions, we multiply by the reciprocal of the denominator:

$$\tan(A + B) = \frac{7}{6} \cdot \frac{3}{2} = \frac{7 \cdot 3}{6 \cdot 2} = \frac{21}{12}$$

6. Simplify the fraction

Finally, we simplify the fraction $\frac{21}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

$$\frac{21}{12} = \frac{21 \div 3}{12 \div 3} = \frac{7}{4}$$