Draw a sketch of the curve \(C\) whose polar equation is \(r=\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2} \pi\).

On the same diagram draw the line \(\theta=\alpha\), where \(0\lt \alpha\lt \frac{1}{2} \pi\).

The region bounded by \(C\) and the line \(\theta=\frac{1}{2} \pi\) is denoted by \(R\). Find the exact value of \(\alpha\) for which the line \(\theta=\alpha\) divides \(R\) into two regions of equal area.