Question

If \(\tan A=\frac{1}{2}\) and \(\tan B=\frac{1}{3}\), then \(A+B=\)

• A. \(30^\circ\)
• B. \(45^\circ\)
• C. \(60^\circ\)
• D. \(90^\circ\)

Hint:

When \(\tan(A+B)=1\), the principal angle is usually \(45^\circ\).

Answer 1

The Correct Option is B

Concept: We use the tangent addition formula: \[ \tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B} \]

Step 1: Substitute the given values. \[ \tan A=\frac{1}{2}, \qquad \tan B=\frac{1}{3} \] \[ \tan(A+B)= \frac{\frac{1}{2}+\frac{1}{3}} {1-\frac{1}{2}\cdot\frac{1}{3}} \]

Step 2: Simplify the numerator. \[ \frac{1}{2}+\frac{1}{3} = \frac{3+2}{6} = \frac{5}{6} \]

Step 3: Simplify the denominator. \[ 1-\frac{1}{2}\cdot\frac{1}{3} = 1-\frac{1}{6} = \frac{5}{6} \]

Step 4: Therefore, \[ \tan(A+B)=\frac{\frac{5}{6}}{\frac{5}{6}}=1 \]

Step 5: Since \[ \tan 45^\circ=1 \] So, \[ A+B=45^\circ \] Therefore, \[ \boxed{45^\circ} \]