Question

If \(\tan\left(\frac{\pi}{4} + \theta\right) = \frac{1}{2}\) then the value of \(\sin 2\theta\) is

• A. $-\frac{1}{5}$
• B. $\frac{2}{5}$
• C. $-\frac{3}{5}$
• D. $\frac{4}{5}$
• E. 1

Hint:

Use $\sin2\theta = \frac{2\tan\theta}{1+\tan^2\theta}$ when $\tan\theta$ is known.

Answer 1

The Correct Option is C

Step 1: Use identity
\[ \tan\left(\frac{\pi}{4} + \theta\right) = \frac{1 + \tan\theta}{1 - \tan\theta} \] \[ \frac{1 + \tan\theta}{1 - \tan\theta} = \frac{1}{2} \]

Step 2: Solve
\[ 2(1 + t) = 1 - t \Rightarrow 2 + 2t = 1 - t \Rightarrow 3t = -1 \Rightarrow t = -\frac{1}{3} \]

Step 3: Find $\sin2\theta$
\[ \sin2\theta = \frac{2t}{1 + t^2} = \frac{2(-1/3)}{1 + 1/9} = \frac{-2/3}{10/9} = -\frac{3}{5} \] Final Conclusion:
Option (C)