Question

The value of \(3tan^{-1}(\frac{1}{2}) = \ldots\)

• A. \(tan^{-1}(\frac{5}{2})\)
• B. \(tan^{-1}(\frac{2}{5})\)
• C. \(cot^{-1}(\frac{11}{2})\)
• D. \(tan^{-1}(\frac{11}{2})\)

Hint:

Use the double angle formula for \(\tan^{-1}\), then add one more \(\tan^{-1}\frac12\).

Answer 1

The Correct Option is D

Step 1: Use the double angle idea step by step
Let $\theta=\tan^{-1}\tfrac12$, so $\tan\theta=\tfrac12$. Then $\tan2\theta=\dfrac{2\cdot\frac12}{1-\frac14}=\dfrac43$.

Step 2: Add one more angle
$\tan3\theta=\dfrac{\tan2\theta+\tan\theta}{1-\tan2\theta\tan\theta}=\dfrac{\frac43+\frac12}{1-\frac23}=\dfrac{11/6}{1/3}=\dfrac{11}{2}$.
Numerically, $3\tan^{-1}\tfrac12=1.3909$ rad and $\tan^{-1}5.5=1.3909$ rad, which agree. So the value is $\tan^{-1}\tfrac{11}{2}$, option (D).

Final Answer:
$3\tan^{-1}\frac12=\tan^{-1}\frac{11}{2}$, option (D). \[ \boxed{\tan^{-1}\dfrac{11}{2}} \]