Discussion
If \(\tan\left(\frac{\pi}{4} + \theta\right) = \frac{1}{2}\) then the value of \(\sin 2\theta\) is
Answer: C
\(\sin 2\theta = \frac{-3}{5}\).
Let \(t = \tan\theta\). Then
\[ \begin{aligned} \tan\left(\frac{\pi}{4} + \theta\right) &= \frac{1 + t}{1 - t} = \frac{1}{2} \\ 2 + 2t &= 1 - t \\ t &= -\frac{1}{3} \end{aligned} \]
\[ \begin{aligned} \sin 2\theta &= \frac{2t}{1 + t^{2}} = \frac{-2/3}{10/9} \\ &= -\frac{3}{5} \end{aligned} \]
\(\frac{4}{5}\) is \(\cos 2\theta = \frac{1 - t^{2}}{1 + t^{2}}\), and dropping the sign of \(t\) gives \(+\frac{3}{5}\), which is not offered.
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